I have always avoided using TT fonts for print, but recently received a template for print that used all True Type fonts in its style sheet. Do you think I'll have problems with the fonts becoming corrupt or any other prolems?
There is no good reason to avoid TrueType fonts for printing. Your choice of font should be based on design considerations and availability from a reliable font foundry, not on whether it is Type 1 or TrueType.
All Adobe PostScript 3 and PDF-based printer RIP products (as well as most Adobe PostScript Level 2-based printer and RIP products beginning in 1995 or so) have natively supported (i.e., no phoney baloney on-the-fly lossy conversions to Type 1) rendering of TrueType fonts including use of the "hints" included in such fonts.
A properly designed and executed TrueType font will render and print every bit as well as a Type 1 font. In terms of crufty amateur-hour fonts downloadable on the web, we've seen as many Type 1 versions of such fonts as we have seen TrueType versions.