Yarm upstream of the road bridge, not sure if both sides but town side is. Preston Park also Bowesfield below the LTA section towards the A66 roadbridge.
Thornaby side opposite Bowesfield round to horseshoe bend.
Some pegs in Stockton town centre below Victoria Bridge but often anglers in these pegs.
I'm no Tees expert, but I believe that there are still a few. I'd suggest that you get confirmation, probably from the local tackle shops, before fishing them. Broken Scar in Darlington and a short stretch at Croft are the two I'm most sure of. Then you might be looking at a stretch at Yarm. There are supposed to be a few more further downstream, Preston Park, Stockton, Bowesfield etc, but I've never fished any of the Tees free stretches.
It's a while since I have fished the Tees but as Chris says Broken Scar in Darlington below the the weir to about 400yds past the Gravel works (on opposite bank) park in the pump house car park was alway free fishing as was the Durham bank in Croft upstream from the road bridge and the bit upstream from the road bridge in Yarm on the south bank. In addition to those the bit along the green in Neasham was allegedly free fishing there were no signs up only fished it a couple of times and no one said anything. but as I said it is a long time since I have fished the Tees so things may have changed
Been a long time since I have fished theses but
Broken Scar used to be for Darlington residents only.
Yarm is town side only. (Small stretch)
This I have fished in the last few years
Opposite Bowsfield (Ingleby / Thornaby side )
From the pipe bridge above the Queen Elizabeth bridge to the A66 bridge.