If I plug my iPhone into another computer (just to charge, no iTunes on the computer) will the owner of that computer have access to my data?

Nope not without your access authentication information which is based on apple ID.

No data access and syns is just possible through itunes and so there is no way without itunes the data cn be accessed.
But yes if you plugged into your iPhone to others PC which does not have iTunes then your iPhone is detected under Scanners & Cameras and from there camer album photos can be accessed and viewed.

If you plug it into a PC without ITunes, the computer will still recognize it as being there. It will(should) prompt an AutoPlay box that asks to Import Pictures and Videos, Open Device to view files and some more depending on your setup. As stated by Karen, the most likely thing they could do is view your photos/videos.

If you're concerned about it. Don't plug it in ;)