for TTs, i prefer to warm-up exclusively on the trainer (though i have done it on roads when I can't travel with a trainer--my warm-up is definitely a bit more scattered in those cases).
i'm usually running a tubular setup with a TT-oriented tire. that means the tire is designed to roll fast at the expense of longevity. the trainer I have with a metal drum is good for many reasons but does start wearing away at the tire right away when tension is applied--if i clean the drum and go for a ride i can see the evidence. i don't care at all about that for indoor training or for most tires, but when a tire is already thin to begin with and when i'd prefer to get a full season out of said tire and re-glue in the off-season when i have more time it matters a bit more.
if i'm noodling on the trainer this also wouldn't make a difference, but i'm generally doing a short-but-focused warm-up where i run through all my zones and put some power down on the trainer.
with the trainer that have a soft contact roller, tire wear is a non-issue even with a thin tire, at least for that season. the seasons before i made the switch i either swapped wheels or wound up having to deal with the extra wear and possibly glue a new tire mid-season. that's not a huge deal, but sometimes it comes at an awkward time. for me, gluing a tire one-off is nearly the same time/effort as gluing 6 new ones.