Nick Chadwick's first-half goal was the difference between the two sides as the Robins failed to progress past this stage for the first time in four years.
Yates said: "It was Tamworth's day and they have done a decent job on us. In the first half, too many of our players were not doing as well as they should have done. That was our downfall today.
"In the second half we threw everything at them and we had chance after chance, leaving ourselves open at the end, which we did. It's massively disappointing and unfortunately we've fallen victim to the FA Cup situation, when anybody can beat anybody.
"We should have had a penalty late on, but credit to Tamworth because they earned that bit of luck at the end."
- Sports & Recreation