Several reactions are performed on a molecule containing a sulfur atom, and this will become a thiol when I'm done. In the literature, a common procedure seems to be to protect it with methyl, and then remove the methyl using t-BuSNa in DMF at elevated temperature in dry conditions.

To practice I used 1,4 dithioanisol. But the reaction failed, not only was there a seemingly slower kinetics (25% conversion) also there was a substitution of SCH3 by SC(CH3)3. This means the thiolate has attacked at the ring instead of the carbon atom.

I do not understand the mechanics here. There is probably a SN2 reaction at work if things go as they should, with the tert butylthiolate ion a huge nucleophile thus a slow one, but it's target methyl will be so bound to the resulting thioether that it won't be replaced any more. However, the t-butylthiolate attacked the ring... this is not normal procedure AFAIK. What can sulfur do here that makes things complex?

Good nucleophiles can participate in what is called S_NAr chemistry, which is what you have observed.  When you use this reaction will you have an aryl-methyl thioether, or an alkyl-methyl thioether?

There is an aryl methyl thioether. This should become an arylthiol and a t-Bu-S-CH3 compound, which is a volatile.

I'd love to know more ways to deal with this, the alternative I have is ammonia and sodium which is all nice but problematic for some groups I'd like to have on the other side.

The good nucleophiles you mention can attack an aryl-X on the ring, I assume you mean the likes of grignard or lithium reageants?