I am most of the way through removing the grout from a shower stall and bathtub surround. My tiles are 4" x 4" and the grout lines are 1/16". It is painstakingly slow work. I started with a Fein oscillating tool and the smallest diamond blade they have, 3/32", was too wide, it chewed up the tile. I am going now with a Dremel and an attachment for grout removal, #568, and a a 1/16" bit #569. It has taken days to get the grout out. If I had more money than time I would have had someone demo out the tile and put in new. That would have been a lot faster. As it is I have time and no income, which is called 'retired', so I chip away.
Some tips are to start with utility knife and make a groove for the dremel bit to slip into, like a guide. Make multiple passes, each time a bit deeper. I got good at controlling the depth, feeling where the grout is soft and thus going deeper, easing up where it is harder, making multiple passes. I make it about 16 squares until the bit breaks. We have broken the job into 30 blocks of 16 squares. One hour per square. One bit per square. That makes 30 bits, at $12 each!
I know with more experience the speed would increase and the bit breakage would decrease.
We started with a vacuum holding person and a dremel operating person, but now have developed a velcro attachment for the vacuum. It is working great!!
Here is a link to the video of the vacuum attachment at work: