Is there any truth to breaking a bat by hitting on the trademark
Yes there is. Traditionally the logo or label is located in a specific orientation to the grain in the wood, leading to the guideline that the logo should face the sky or the ground ("logo up or logo down") meaning you would be hitting the ball at 90° to the position of the logo.
As one manufacturer puts it:
There is a proper side to hit on, so be sure to also see the answer to the next question.
...[by orienting the label up or down] you'll be making contact with the area on the "side" of the bat where the grains are layered---the strongest "side" of the bat.
Unfortunately, and as usual, it's not quite as simple as that because more than one species is used to make baseball bats and they have quite different structures. Also there is more than one type of breakage/failure (in the handle area or the barrel). More info here on WoodBats.Org.
does burning a trademark into the barrel weaken the bat?
If it's placed in the traditional location, I doubt it. It's at a right angle to the primary forces the bat is (should be) subjected to, so it should normally have no effect.
A burned-in logo would normally only involve the surface anyway. It is possible that if you burn very deeply it might compromise the wood if the logo is placed atypically, but I think that you'd have to have both conditions met for it to affect a bat's strength — so both burned far too deep and in the wrong orientation to the grain.