According to grodenhiATgmailDOTcom :

The wainscoting I was planning on installing is the bead board type
(at Lowes they sell 3/16" white beadboard).

Is this tongue-and-groove (often around 3 1/4" wide) or sheet material
(often pressboard of some sort)?

My stuff was 3 1/4" x 3/16 x 8'.

So it looks like my best
bet is to remove the tiles and either replace the drywall (if damaged)
or nail this over where tile used to be. I really hope to not have to
replace the wall board as the walls I want to do have baseboard
heating and I don't want to have to remove that to. I was really
hoping for something along the lines of a weekend project.

You're exceedingly unlikely to get the tile off the drywall without
considerable damage. It'll take vastly longer if you try to save
the drywall. You can probably rip the drywall/tile off in less than
an hour, and it won't take much longer than that to put up new
drywall, because behind the wainscotting you don't have to "finish mud"
it. (you still should tape and first pass the joints).

With a circular saw, plywood would take only a bit longer to install,
and you don't have to wait for the mud to dry.
--
Chris Lewis,

Age and Treachery will Triumph over Youth and Skill
It's not just anyone who gets a Starship Cruiser class named after them.