I am assuming you want radials, but you never did say what you wanted out of your tires, performance or originality.
I am an original size tire kind of guy, so that is what I will write about.
My friends in high school that had TR3s all had 165-15 Michelin X radials on them and I believe that was the original radial option for TR3s. Coker also lists that as the correct size radial, but they do not list all the 165-15 tires they sell when you search their using tire size on their website.
Between Coker and Universal tire they sell at least six different 165-15 tires.
For black walls, on the low end is Firestone tires at $80 each from Universal and Coker. You probably get what you pay for their. Next is Vredestein Sprint Classic (narrow tread) 165-15 tires at $119 from both. Michelin 165-15 XZX tires are next for $161 each from Coker. The high end is the Michelin XAS 165-15 for $295 each. On the Coker website most of the tires do not come up when you type in the size 165, All Aspect, 15. You need to select Brands along the left side then select each brand they style then it will show the sizes.
I'm not, but if you are into white wall tires for your TR3, then Coker makes the Coker Classic with 2 1/4 wide whitewalls for $160 each and they also sell a tire called the American Classic with 2 1/4 white walls for $192 each.
I don't understand the one comment above. Just because a tire supplier or manufacturer does not list a specific car maker does not mean they don't have the correct tires for your car.
Vila
1933 Chevrolet
1962 Triumph TR4
1984 BMW 633 CSi