Hi there, I've been walking up on my toes since I could walk, to the point that I feel uncomfortable to walk the normal way (without shoes on. With shoes I walk normally). I spend a lot of time barefoot, and only walk with my heels down when in company with someone I don't know well, or when walking very slowly.
The short answer is, the exact potential problems can depend on why you walk on your toes. There can be underlying medical conditions (such as shortened tendons, bone blocks, and various foot issues) which cause it and would require medical attention. It can also be an early marker for autism (again, hi, worth noting I am on the autistic spectrum, but very high functioning), and is more common in men than in women.
It can also just be that some people prefer to do it. This is called idiopathic toe walking. A lot of children go through a phase of toe-walking and then grow out of it, but some don't.
I've done a lot of reading about this over the years to try and work out if it is something that I should worry about, or try and fix, and the simple answer is.... Probably not as long as you are careful about it and don't do it constantly. Common issues from constant toe walking can include tight calf muscles and lack of ankle movement, as well as occasionally other leg joint issues, an increased chance of slipping and falling, etc. Personally, I am quite prone to pulling on the tendons around the outside edge of the ankle. You can help avoid these issues with stretches and just... walking flat footed sometimes.
So while it isn't something you have to stop doing completely, I would recommend you don't do it all the time.
(I am not a doctor but I have had a life of walking like this and done some research, if you are genuinely concerned then talk to your medical practitioner)