Hi Mike,
First check that all the steering linkage is free, lubed, working properly and that bushings aren't sloppy. The tie-rod ends sold today mostly come without grease fittings, might look for some that have them (I think TRF offers them for TR4, not sure what's avail. for TR3). The vertical link/trunnions should be checked for free movement, too.
The best way to check these things is to remove the shocks and springs, then operate the steering mechanism by hand with the car up on jack stands. With the springs and shocks off, it's possible to test at different deflections of the suspension, see if all is working correctly throughout the swing of the axle and there are no binding points.
You might want to take the car to a local alignment professional and have them look at it. TR3 steering is heavier to begin with, of course. Modern radial tires can add to the problem, since the car was originally designed for the bias belted tires that were most common at the time. Perhaps the alignment shop can do some tweaking to improve matters.
If caster and/or camber and/or toe is off it can make for heavier steering. They should check those and can adjust them on most cars. TR2/3/4 have no obvious caster/camber adjustment points, but the vertical links can be "adjusted" by bending them (or can get out of adjustment if bent). There are also adjustable top fulcrums available, from Revington TR, but these are usually used to set up negative camber for racing and such. They might not be useful to you, but I've posted photos of these installed on my TR4 at
https://www.triumphowners.com/640 if you want to see them. Note that these fulcrums are being used on a TR4 with 3 degrees of caster. I'm not certain if 0 degree caster type fulcrums are available (0 degree was used on TR2 through very early TR4).
Look for an alignment/frame/suspension shop that is a step above the typical, perhaps someone who specializes in older cars. They can probably give you better advice specific to your car, after looking everything over carefully.
There are a few other things that might be done. It might help to simply run the tires harder, the tire pressures recommended by the manuals are also most likely for "old technology" tires. You could try this carefully, or look for advice from the tire manufacturer or the local alignment shop.
Hope this helps!
Alan Myers
San Jose, Calif.
'62 TR4 CT17602L