No problems. I once had a certain factory rim that developed a leak, in the rim inside area. The tire shop said I needed a new rim, which would have been hard to find. I told him to put a tube in. He had this blank expression on his face, that to me, appeared to say "I am the stupidest person, shy didnt I think of that". Ran it that way for years, no problems.
Now, most tire shops will probably try to discourage you from doing that, because of the additional work and time to mount them. It can just be thrown on the tire machine and roughly mounted, it takes a little care to not pinch the tube. Sorry for being longwinded.
I had a Morris Minor Traveller while in England and had new tyres put on and went somewhere with my daughter and had four flats. After I gave her a piggy back ride for a mile and a half I told the mechanic to put tubes in. Did not have any more problems with the car or mechanic after that.
If you are thinking that putting a tube in a tire in order to solve a leaking problem - well...ah...
You should make sure you haven't used the tire too much when it was underinflated. Underinflated tires eventually fail. And a tire never forgets it was underinflated, so replacing the tire is the safest action you can take.
Don't forget to solve any structural problems that may exist. A puncture not only makes the tire not air tight, but it also damages the tire's structure. Patches have some reinforcement to add that structure back.
If the problem is the wheel, it might be better to clearcoat the wheel.
BTW, some wheels have some unusual contours and may overly stretch the tube, causing the tube to eventually fail.
On the whole, using a tube is not a great idea.
TR3 wire wheels only work with tubes. I have to run tubeless tires with tubes since 1989 as the tube types are hard to find. Just make sure the wheels are smooth inside or use a rubber liner to protect the tube. I do both. Rubber liners available through Moss or Vintage Triumph.
If it's for a slow around town driver it's maybe ok. A lot of friction is built up between the tube and a flexable radial tire that will result in tire failure in short order at highway speeds. Check with any of the major tire web site and ask to question. Years ago it was considered DANGEROUS. ed
For motorcycles anyway, a tube or tube-tubeless tire has a smoother inside finish to protect the tube. Somewhere I read that you should drop down one speed grade (V to H, H to S, etc.) if you put a tube in a tube/tubeless tire.
For a car, can you even get tube-type tires anymore except for say wire wheel British sports cars?
If you have a tire that is leaking air, rather than go to a tube maybe better see if the rim/bead seat is dirty or corroded, or the tire is damaged. Except for agricultural speeds I would not try to "fix" a leaker with a tube...to be blunt a guy can get killed in a blowout...it ain't worth it!
When I was a kid my Dad would run cheapie recap bias-ply tires with tubes in them on his old half-ton GMC pickup...sometimes hauling a full ton of scrap iron in it...70mph in Atlanta traffic on I -85...He now heartily agrees with me that this was as dumb a stunt as anything I did in my teens...Truck is mine now and the improvement in putting a decent set of radial tires on it was dramatic to say the least.
Whilst it is possible to put tubes in some tyres, it is not advised as modern tubeless tyres are not really built to take tubes for the reasons given. The other thing is that you need to still repair the tyre as the rough edges will rip the tube plus any moisture that gets into the hole will rust the radial belts. Not good.