How to deal with a colleague who won't accept they're wrong

Use impersonal language

Hopefully this is obvious, but make sure that your language focuses on the ideas rather than the people. Rather than

I don't think you're grasping the subtleties of the situation

try

I think that there are some extra complications that need consideration in this case

Ask questions

Rather than taking a confrontational approach, try to adopt the role of the inquisitive student. Rather than

There is evidence in publications X and Y that directly contradicts what you are saying

try

What are your reasons for disagreeing with publications X and Y?

or

I'm having trouble understanding why the conclusions from publications X and Y do not apply here. Can you elaborate?

This will hopefully give your colleague a graceful way out without having to turn it into a matter of "you're wrong and I'm right". As a bonus, it also gives you a way out if it turns out you were wrong after all (however unlikely that is) or that you misunderstood your colleague's point. But as @R.M. and @CaptainEmacs have pointed out in the comments, take care not to overdo it, or you could come across as either clueless, or patronising.

Make someone else the scapegoat

Rather than voicing your concerns directly, put them in the mouth of a hypothetical reviewer. Rather than

I think there are these reasons for rejecting that argument

try

I think that a reviewer could object to that argument for [reasons]. I think we need to anticipate this using [different argument].

User2390246's answer offers great advice if you need the other person to confess that they are wrong. I just want to add that this is not always necessary. In that case, the best way to deal with your stubborn interlocutor is to agree to disagree.

I can think of three situations in which it is desirable to have the other person concede an error:

1. Improving your interlocutor's knowledge: You are essentially providing a free service to the other person by pointing out their mistakes. It is their decision whether to accept this service or not. If they don't, agree to disagree. You certainly can't be expected to make someone else happy against their wish.
2. Improving collective knowledge: If you are working together, there are certain issues on which you have to come to an agreement -- at the latest, when it comes to writing down research findings and their interpretation. Other issues may be tangential and can be allowed to rest, or they may allow different interpretations, which may lend themselves to be framed as a discussion of the results. However, if you are not collaborating on the same project, there is even less reason to agree.
3. Status signaling: Insisting on a wrong (or weak) point can be perceived as being necessary to protect social status. A senior person who feels their status is already precarious may feel "called out" by a junior person who points out their mistake. The junior person may want to consider if they want to go through the trouble of holding their ground, or if they wouldn't rather play along in the status game, while distancing themselves internally. ("I know I'm right, but if you need to save face, that's fine with me.") Of course this is not an either/or question but a matter of degree.

If you don't really need to protect your status, and if by insisting on the truth you neither realistically improve your interlocutor's nor your collective knowledge, it is best to agree to disagree.

Given that the top voted comment states that "People who put belief and/or pride over evidence has[sic] no place in STEM," and other answers have given diplomatic approaches for bridging the gap, I would like to address why such a situation might occur. In particular, this situation is reminiscent of Kuhn's Structure of Scientific Revolutions.

While that book did spawn generations of middle-management rambling about "paradigm shifts" and "thinking outside the box," it contains a very fundamental insight: as much as we would love to believe that we are purely logical researchers driven by objectivity, we are (un)fortunately loosely contained sacks of meat. We are still governed by heuristics and social dynamics, even if we try to place a higher emphasis on logical reasoning. The history of science does not read "Darwin published On the Origin of Species and nobody ever questioned natural selection ever again" or "Einstein wrote a couple papers in 1905 and every physicist immediately agreed with his views and threw away their old texts."

However, Kuhn does give you some solace: eventually the old guard will die out and the new theory will be regarded as the dominant paradigm. Unfortunately, you will also age until you in turn become the old guard, arguing about why the new upstart theory can't possibly be correct.