as already stated, the better the traction, the better the 60' and the more stuff you're going to break.
MT DRs are the stickiest you can buy....followed by the BFGs....followed by the nittos. personally, i run nitto DRs on the street and BFG DRs @ the strip. running BFG or MT DRs on the street will lead to a VERY short lifespan (3k miles? if that...).
what you break depends on how you launch...and what trans you have. the first thing to go in a stock auto trans'd car is the auto trans- they can't deal w/the load. in a manual trans'd car, it all depends on your launch technique. slipping the clutch will be far kinder to the trans and the rest of the drivetrain/suspension than a clutch drop....but it will also burn up you clutch rather quickly.
the manual transmissions aren't perfect. the W58 has weak gears and the R154 has the thrust washer issue. your best bet is to baby it if you have the W58 and to run a GOOD oil if you have the R154. i personally have over 200 passes on my R154 and have no issues, thanks mostly to redline heavy shockproof and no speed-shifting.
speed-shifting (flat foot shifting, shifting w/o the clutch, etc.) is a bad idea. the transmissions are not up to it. don't do it unless you like to change the transmission.
the rear subframe is something to be concerned with. if the diff mounting bolts/nuts are even the slightest bit loose, the diff will rock around and beat your subframe to a pulp. the '89+ braces help to hold the subframe together, but the best thing to do is weld the heck out of it.
as already stated, the "strut rods" (trailing arms...) are fairly weak and will bend if your car isn't properly aligned, etc. i personally haven't bent one yet.
the clutch pack diff is plenty strong. the torsen, on the other hand, will blow up if you even think of going to the strip.
no concerns with the driveshaft, as IJ said, it's plenty strong as well...
that's just my $5.....take it FWIW, YMMV, etc.