Describe your knee pain in more detail if you will, please. Typically a "burning" sensation is associated with the doctors favorite go-to diagnosis, Patellofemoral pain syndrome, PFS, whatever you want to call it.... Most people don't care what the doc wants to call it. They want to know how to treat it and then prevent it.
Of course it is recommended that you seek a referral for an orthopedic specialist in your area for an accurate diagnosis. It appears you've seen a doctor for this issue, but you didn't say if it was a primary care physician or ortho specialist. I understand some people do not have access to health care or cannot get the help or answers they need from their providers. With that being said, I am in no way providing you a diagnosis. Or prescribing you therapy techniques. Leave that up to your doctor/s or a really good physical therapist. However, I would like to share some information with you, so you can have a better understanding, and maybe some of this will help you.. I hope it does, and I hope you recover soon.
Unless you have had a traumatic injury to the knee there usually will be no suspicion of any internal derangement. Meaning the likelihood of a torn ligament, damaged cartilage (such as the meniscus), torn tendon or broken bone is very slim. A burning sensation that comes out of nowhere could easily be a tendonitis type issue. Which can sometimes be a pain to get rid of. Im talking months of therapy and rest possibly.
If your pain is located close to your kneecap on either side, you likely have a tracking issue. Meaning you probably have an imbalance in your quadriceps muscles leading to your kneecap pulling one direction more vs the other and is likely to be a source of pain. Think about it like this. Your kneecap fits in a groove and if your inner thigh muscle is overpowering your outer thigh muscle it can pull on that kneecap and over time doing repetitive movements especially under a load can cause some wear and tear and lead to inflammation. Fixing the imbalance of muscle strength throughout the quadriceps muscles can help. It will take time but it can help. That combined with resting and any other therapy associated with this issue should help if that is what is causing the pain. I know you said your strength is good and I believe you that your strength is good so it is confusing right? Doc says strengthen your quads but you know you lift double what the average guy does so You're wondering... Yeah ok..what are you talking about...nerd...lol.. It may not be strength but as I said before an imbalance. This muscle overpowering this other muscle here. Balancing them out may help. An ortho doc or physical therapist can run some tests to determine what's going on. Who knows you may not even have an imbalance or weakness. The human body is crazy. There's so much we still don't know. So much that is unexplainable. We know less than we think we do. It can be frustrating
If your pain is located on the lateral (outer) side of the knee it's possible that you have an Illiotibial band (IT band) issue. The IT band essentially runs from your hip to the knee. The lower attachment site is controversial and "stretching" therapy of this tendon is controversial as well. But anyway basically it runs from your hip to your knee and is a very thick band. Imbalances or weakenesses in the often neglected glute and hamstring muscles can cause tightness and pain in the IT band which usually flares up at the lateral (outer side) joint line of the knee. If this is your issue and what you're diagnosed with they will likely have you foam rolling your it band and "stretching" it or massaging it or maybe they will do some dry needling or a-stym or graston stuff on you. IT band syndrome sucks. And can take forever to recover. Bleh. Resting is key for this one. Graston therapy works wonders on it.

therapy will probably consist of hamstring and glute strengthening exercises until your so overwhelmed and annoyed you just wanna leave therapy and never go back.. Or they'll try to get you to strengthen the inner or outer quad muscles BLAH blah. Hopefully you find a good physical therapist who knows what they're talking about. Otherwise you'll be doing really pointless exercises. And.. Don't be afraid to speak up and tell them something isn't working or you're not feeling a stretch.
Again. Talk to your physician/ortho/PT for guidance. Don't take what I say for gospel. Based on what you described and what your doc diagnosed you with. These two syndromes are most likely suspected. There could be other contributing factors as well. The more info/background you can give on the issue the better you can be helped..
Sorry for the extremely long post...But if you took the time to read it hopefully it helps you understand better what may be going on so you can fix the problem. If it's a tendonitis, resting and therapy will get you better. And fixing the muscular imbalances will help prevent it from coming back. And it takes time. There's no overnight fix. NSAIDs (ibuprofen naproxen whatever) can help along with heat/ice. But you won't be better right away. Sometimes tendonitis will hang around for months and one day it's gone. the Human body is amazing I tell you.
Good luck with your knee issue. Get well soon.