[epistemic status: mainly based on Lucio Russo's papers, and on my personal research on the sources]
TL;DR: Although it is widely believed that Aristarchus' heliocentric theory was rejected by his contemporaries, we do not have much evidence supporting this belief. We have too few surviving texts to decide, but some of them seem to hint that many Hellenistic astronomers may actually have accepted heliocentrism.
The aim of this post is to provide evidence that heliocentrim was a respected theory (and possibly even the expert consensus) among astronomers in the third and second century BCE. In the first section, we will see that we have no reason to believe that Aristarchus was condemned for his theory. In the second section, I list some ancient sources that show that heliocentrism had at least some followers among ancient scientists. Finally, in the third section we will see a possible way in which Hellenistic scientists could may have convinced themselves that heliocentism is true.
I. Aristarchus was never accused of impiety
The belief that Aristarchus was condemned originated from the one of the most puzzling dialogues of Plutarch, On the face which appears in the orb of the Moon. We do not have the full text of this dialogue, but in the beginning of the surviving part (Plut. De Faciae 6) it is written (in the translation of Harold Cherniss):
Thereupon Lucius laughed and said: ‘Oh, sir, just don't bring suit against us for impiety as Cleanthes thought that the Greeks ought to lay an action for impiety against Aristarchus the Samian on the ground that he was disturbing the hearth of the universe because he sought to save (the) phenomena by assuming that the heaven is at rest while the earth is revolving along the ecliptic and at the same time is rotating about its own axis.
Cherniss translated "Cleanthes accused Aristarchus" based on the accepted greek text, which put Cleanthes in the nominative case (Κλεάνθης), as the subject of the sentence, and Aristarchus in the accusative case (Ἀρίσταρχον), as -no pun intended- the receiver of the accusation. Except that this is not what is written in the original manuscript. The original text of On the face of the moon survives in two codicis (Parisinus B and Parisinus E), and in both of them Aristarchus is in the nominative case (Ἀρίσταρχος) and Cleanthes is in the nominative (Κλεάνθη). So, if we read the original, it is Aristarchus who accused Cleonthes, and not the reverse. This makes also much more sense in the context of the dialogue.
The text was amended in the XVII century by Gilles Ménage. Early modern philologists, influenced by the trial of Galileo, were probably puzzled by a text who said that Aristarchus accused someone; so they decided to solve the confusion by correcting what the source said.
II. Aristarchus was not alone
Nearly every scientific text from the Hellenistic age has been lost . Since we can read Ptolemy's Almagest, we know that Ptolemy was a geocentrist. All the claims that "heliocentrism was dismissed in the ancient world" are essentially based on the interpolation that, since Ptolemy was a geocentrist, everyone else before must have been a geocentrist. But actually we have little information on what astronomers believed in the four centuries between Aristarchus and Ptolemy.
This is all but an homogeneous period of time: we can divide it in a phase of great scientific activity, in which lived scientists of the league of Archimedes, Ipparchus of Nicaea and Apollonius of Perga; and in in which astronomic research was largely discontinued. This is evident from the following graph, which plots the dated astronomical observations quoted by Ptolemy in the Almagest .
Since we can not read the original sources, we have to rely on later writers who quote them. In the II century, the Skeptic philosopher Sextus Empiricus wrote Adversus mathematics, in which he criticises all kinds of academical knowledge (due to the loss of almost all the original scientific sources from that period, his work is -somewhat ironically- a precious source of information on ancient mathematics and science).
In Adversus Physicos, II, §174, Sextus Empiricus attributes the heliocentric hypothesis to "Aristarchus and its followers" (οἱ περὶ Ἀρίσταρχον). This means, at least, that Aristarchus had some followers.
Can we name some of these followers?
For start, we know that Archimedes was a heliocentrist. In The Sand Reckoner, the letter in which Archimedes invented the exponential notation to estimate an upper bound for the number of grains of sand that would fit in the Solar System, Archimedes explicitly employs in the calculation the heliocentric system of Aristarchus. Furthermore, it is known that Archimedes built a mechanical planetarium. Cicero (De re publica, I, xiv, 22) praised Archimedes' planetarium writing that he managed to reproduce all the motions of the planets with only one "conversio" (only one joint?).
Plutarchus (Platonicae quaestiones, VIII, i) gives us the name of another ancient supporter of heliocentrism:
Does the earth move like the sun, moon, and five planets, which for their motions he calls organs or instruments of time? Or is the earth fixed to the axis of the universe; yet not so built as to remain immovable, but to turn and wheel about, as Aristarchus and Seleucus have shown since; Aristarchus only supposing it, Seleucus positively asserting it?
This is interesting, because Plutarchus apparently believed that Seulecus of Seulekia not only accepted Aristarchus' system, but also proved it ("καὶ ἀποφαινόμενος", which in the above translation is rendered as "positively asserting"). We will see in the next section what his argument could have been.
Several passages from Roman writers, like Lucretius (De Rerum Natura, IV, 387-390) and Senecas (Naturales quaestiones, VII, xxv, 6-7), describe the retrogradation of planets as an apparent phenomenon arising from the combined motion of the Earth and of the other planet. Lucretius poetically compares it with the apparent motion of hills and plains when seen from a ship. Senecas also explicitly refutes the idea that planets could actually stop and invert their motion (like they do as observed from Earth), because otherwise they "would fall on each other" (alia aliis incident).
Finally, the prominent Roman enciclopedist Pliny the Elder supported Heliocentrism (Naturalis Historia, II, 8), through he did so with a completely wrong argument. This is not strange, since Pliny had an habit of reading correct results and reporting them with fanciful justifications. To make one funny example, Pliny stated that the hexagonal tiling is optimal for honeycombs, because each paw of the bee builds a side (whereas we know from Pappus that the Greeks understood the true reason for which hexagonal tilings are "optimal").
III. The case for Seulecus: Heliocentrism can be proven with tides (not only with parallax)
Contrarily to what is often said, measuring tiny parallaxes is not the only way to confirm experimentally that heliocentrism is true. Ancient astronomers could not detect stellar parallaxis, but neither could Newton or Laplace, or anyone before the XIX century.
An alternative, and arguably easier route to prove heliocentrism, is to understand the dynamical origin of tides.
Today we know that tides result from the composition of gravitational attraction and centrifugal force. Let us read again from Plutarch's On the face of the moon:
Yet the moon is saved from falling by its very motion and the rapidity of its revolution, just as missiles placed in slings are kept from falling by being whirled around in a circle. For each thing is governed by its natural motion unless it be diverted by something else. That is why the moon is not governed by its weight: the weight has its influence frustrated by the rotatory motion.
The analogy with the sling has been seen, by many historians of science, as a correct qualitative explanation of the fact that the orbital circular motion results from the composition of inertia and of a centripetal force. It was repeated by later authors (including some Bizantine archbisop), even when its original sense had been lost. People in the early modern era did sense that this passage was somehow important (for example, it was widely cited by Newton in his early essays). A more technical description of the same idea (i.e., that circular motion arises from a centripetal force) can be read for example in the pseudo-Aristole's Mechanics.
Now, to understand in a qualitative way why tides work the way they do, we can extend the analogy with the sling rotation a bit further and notice that, if we place in the cradle of the sling an elastic object, it deforms and becomes radially elongated.
Since the Earth revolves around the sum, we analogously expect the surface of the oceans to elongate in the direction of the sun. This effect interferes with the gravitational attraction of the moon: when the Sun and the Moon are aligned, the amplitude of the tides is maximised; while when they are at a right angle, we have the lowest tide.
Could Hellenistic scientists understand that?
The Greeks did have a predictive theory of tides, which explained their monthly cycle by the composition of effects of the Sun and the Moon. We can read about this theory, for example, in the sixth of the Answers to Chosroes, written by the Neoacademic philosopher Priscian of Lydia during his exile at the court of the King of Persia .
Seleucus of Seulecia, the scientist to whom Plutarcus attributes the "proof" of Heliocentrism, is remebered by the surviving sources as an expert of tides (for istance, according to Strabo, Hipparchus cited Seulecus' work when he argued that, because of the difference in tide levels between the Atlantic and the Indian Ocean, they had to be separated by some unknown continent).
Galileo, who surely read his Plutarch, devotes the fourth -and last- day of his Dialogue Concerning the Two Chief World Systems to the discussion of tides, apparently sure that they provide the definitive proof of heliocentrism (and it would have been the definitive proof of heliocentrism, had not Galileo got the explanation completely wrong). Sixty years later, the explanation of tides was the main success in Newton's Principia Mathematica.
To sum up, according to Plutarchus, the Heliocentric theory was "proven" by someone who happened to have as main interest the only phenomenon (bar stellar parallaxis) which could reasonably have been used by ancient scientists to prove heliocentrism.
 In the previous lines, Pharnaces told Lucius that his position was absurd and that he is "turning the universe upside down". Lucius replies that his position is instead much more natural than Pharnaces' one, and he compares Pharnaces to Aristarchos, who also accused Cleanthes despite the fact that his own idea was much more counterintuitive.
 The exceptions Aristarchus' On the Sizes and Distances of the Sun and Moon, and Ipparchus commentary on Aratus' Phaenomena. The latter, hardly an important work, was probably saved due to the popularity of Aratus' poem.
 It is often claimed, even by the Encyclopedia Britannica, that Hipparchus was a geocentrist. But apparently the argument is "Ptolemy cited Hipparchus on an unrelated subject, and Ptolemy was geocentrist, therefore Hipparchus was geocentrist".
 See O. Pedersen, A survey of the Almagest, 2011, Appendix A.
 Quoting a lost work by Posidonius of Apamea, Priscianus explains that at the full moon and at the new moon the effects of the Sun and of the Moon are summed, resulting in the highest amplitude of tides, while at the quarters of Moon the effects of the Sun and the Moon are opposite, resulting in the lowest amplitude. He also correctly says that the highest tides happen at the equinoxes, and that the effect of the Moon is greater than the effect of the Sun.