I would like to take a shot at this, here is one perspective (take it with a grain of salt!):
The TL;DR is perhaps: Yes tongues are for both believers and unbelievers in various ways, but the emphasis of 1 Corinthians 14:20-25 would be that they are 'for unbelievers' (as a rebuke). Nevertheless, through this passage and others, we can say that as they progress (hopefully) to becoming a believer, they may experience tongues (and interpretation) as a blessing. (an interesting turnaround!)
Longer explanation:
We know that in 1 Corinthians 14:21, where tongues are mentioned, it is apparently a reference to Isaiah 28:11.
I would like to also point out that in 1 Corinthians 14:20, Paul gives an interesting take on a kind of "childlike faith", if you will:
Brothers and sisters, stop thinking like children. In regard to evil be infants, but in your thinking be adults.
... and this kind of parallels with a slightly earlier Isaiah verse, in Isaiah 28:9-10:
9 “Who is it he is trying to teach? To whom is he explaining his message? To children weaned from their milk, to those just taken from the breast? 10 For it is: Do this, do that, a rule for this, a rule for that; a little here, a little there.”
But here, in Isaiah, 'the people' have gotten it wrong it would seem. They seem to be unreceptive (and unwise) regarding the message of the prophet, comparing it to a childish message.
So the rebuke of the Lord, is that they then receive 'tongues' -- a stranger message than the simple one first received.
So in a sense, I think these stubborn/complaining Israelites, are essentially unbelievers, who must now hear a message of tongues.
This then fulfills the message in 1 Corinthians 14:22 that "tongues... are a sign... for unbelievers". (yes, even Jews here are being called unbelievers.)
However, if these same unbelievers are to truly hear a message of salvation, it must ultimately be understood and received by them, in a way they can understand (in English! or whatever their native tongue is), and most importantly of course, embraced by the heart.
In this sense, 'prophecy' (a simpler/clear message from the Lord, like a sermon) is also "for unbelievers", to allow them to be converted to a believer.
This of course seems to be what we see in 1 Corinthians 14:25:
...So they will fall down and worship God, exclaiming, “God is really among you!”
So overall, I think the 1 Corinthians 14:20-25 passage is kind of honing in on the unbeliever, as it draws out these truths. And this emphasis is perhaps in contrast with the rest of the chapter, which talks more about believers and the church. But anyway, to complete the picture, concerning believers specifically, we can say 'yes, tongues are for them (spoken by them, and yielding a blessing for them)' and also, 'yes prophecy is for believers as well (or anyone who has ears to receive the Lord and His message!)'.