I'm not sure what the "Torso Compression Tank" is, but I am familiar with Underworks (Manshape retailer?).
It is my understanding that some compression tops have more to do with working out, or are designed for posture or back trouble. What many of us began using as "binders" were actually gynocomastic or post-surgical compression tops for men. Sports compression tanks are not meant for, and will not, suppress and/or conceal excess breast tissue.
It appears that Underworks has overhauled their website, as now, there is not just a page to items they discovered many transmen were using, there is actually like a website now, f2mbinders.com - it is still Underworks. I have always ordered #997 Double Panel Compression Shirt, however, it appears that Underworks (f2mbinders) has changed some model numbers, as this number returns an item that I have never had before.
You can find more information about Underworks and other sites and binders in the thread
As for putting your binder in the dryer, unless something has changed in the last year or two, Underworks advises that the compression shirts not be put in the dryer, and you should also not use bleach or fabric softeners during washing. I do not know the reasons for these care instructions, but I have adhered to them religiously. I have one binder that is almost 10 years old - and I wear mine 365 days of the year.
If your binder is too tight, you could possibly cause issues with circulation and breathing. It's best to order your shirts according to Underworks' sizing charts. You might also have to try different models.
-Michael