If I have some sheet music for tenor sax in A major, is that a transposed A major, or concert A major?
It depends on what you mean by "I have some sheet music for tenor sax in A major." If you are playing a piece in a major key and the saxophone part has three sharps in the key signature then the piece is in (concert) G major, so I guess you could say that the saxophone part is in "written A major" or "transposed A major." If your band is playing a piece in A major then the saxophone part will be written in B major.
If the sheet music for tenor is concert C ...
This is very unlikely. If the sheet music is written in concert pitch then it wasn't written for tenor sax. In that case, you'll have to transpose it yourself. Or, if the piece is in C then the tenor saxophone part will be written in D.
(There is an exception to the above statement "If the sheet music is written in concert pitch then it wasn't written for tenor sax." Sometimes, the conductor's score will have all the transposing instruments written in their sounding pitch, often called a concert score. If someone created a set of parts by cutting and pasting from a copy of such a score -- and by "cut and paste" I mean "using scissors and transparent tape or paper glue" -- then the result would include a part that is specifically for tenor sax but written in concert pitch. In this case you would also need to transpose the part yourself, of course.)
... and it's the only instrument transposing, doesn't that mess up the chord progression?
Not if it's done correctly. But the explanation depends on what you mean by "sheet music for tenor is concert C." I suppose you mean that you're playing tenor sax in a group and everyone is playing from the same lead sheet. This means that you have to transpose the part yourself. When you see a G, you have to play an A. If you are using the chord symbols to decide what notes to play in an improvisation, you have to transpose the chord symbol. When you see "Fm7" you have to play something that fits with a Gm7 chord.
On the other hand, if it's a band arrangement for a piece in concert C then the tenor sax part will be written in D and won't normally have chord symbols. When the tenor sax is playing the fifth of a concert A minor chord, the note will be written as F♯. This won't mess up the chord because all the non-transposing instruments will have A, C, or E.
Don't I then play A major transposed to Bb sax over an A major chord progression?
If the piece is in concert C then the tenor sax will be written in D major. The alto (and baritone) sax will be written in A major.