Small sample size (92 participants), not replicated, and full of potential confounds (maybe the research assistant was extra enthusiastic when handing out envelopes, or maybe the person soliciting noticed who had what and also varied their reaction). That there was an effect at all was dubious, and privileging this hypothesis seems unwarranted. It's plausible, but this study is very, very weak evidence.
Since I'm not very familiar with the Bible, I had to look up the quote in Wikipedia (the full quote being "But when you do merciful deeds, don’t let your left hand know what your right hand does"), and even there it suggests multiple possible interpretations for Matthew 6:3. Can you spell out what you think the quote means, and why this study validates it?
Here is the quote in context:
So when you give to the poor, do not sound a trumpet before you, as the hypocrites do in the synagogues and in the streets, so that they may be honored by men. Truly I say to you, they have their reward in full. But when you give to the poor, do not let your left hand know what your right hand is doing, so that your giving will be in secret; and your Father who sees what is done in secret will reward you.
In my Christian upbringing, this was interpreted as colorful imagery making the point that you should not let attention motivate you to do good deeds, and that in fact you should make an effort to not get attention from your good deeds by doing them in secret. The reward from 'your Father' may additionally be interpreted as that which a truly good person values - the actual goodness caused by the action.
In the study, people who have an obvious signal that they have done good feel satisfied, and thus "have their reward in full," and don't donate as much as the people with the envelope. I think the OP is implying that by intentionally avoiding signalling good deeds, we will end up doing more actual good - thus validating the advice of the scripture.
So, here's the study¹:
It's veterans' day in Canada. As any good Canadian knows, you're supposed to wear a poppy to show you support the veterans (it has something to do with Flanders Field). As people enter a concourse on the university, a person there does one of three things: gives them a poppy to wear on their clothes; gives them an envelope to carry and tells them (truthfully) that there's a poppy inside; or gives them nothing. Then, after they've crossed the concourse, another person asks them if they want to put donations in a box to support Canadian war veterans.
Who do you think gives the most?
...
If you guessed that it's the people who got the poppy inside the envelope, you're right. 78% of them gave, for an overall average donation of $0.86. That compares to 58% of the people wearing the poppy, for an average donation of $0.34; and 56% of those with no poppy, for an average of $0.15.
Why did the envelope holders give the most? Unlike the no-poppy group, they had been reminded of the expectation of supporting veterans; but unlike the poppy-wearers, they hadn't been given an easy, cost-free means of demonstrating their support.
I think this research has obvious applications, both to fundraising and to self-hacking. It also validates the bible quote (Matthew 6:3) which is the title of this article.
¹ The Nature of Slacktivism: How the Social Observability of an Initial Act of Token Support Affects Subsequent Prosocial Action; K Kristofferson, K White, J Peloza - Journal of Consumer Research, 2014