I've made close to 25 t-shirt quilts. I usually will cut very roughly the parts I pant to use away from the neck and sleeves. then I put the interfacing on. I will then cut more precisely down to the size I want for the block. T-shirts will stretch on you so you need the interfacing on to stablelize before you cut to your precise cuts.
Also, be sure to read the instructions with your interfacing--I use a light weight Pellon or recently found a 54" wide woven fuseable interfacing that is often used in tailoring. Some will instruct that you use a high heat, others a medium heat. I find that sometimes I am given t-shirts that are a poly/cotton blend, or even sports jerseys that are all poly--you need to be sure to use an interfacing that works with the lower temps needed for them. Good luck!