Look at the City of Nineveh; they prayed that God would spare their city and He did.
I wasn't going to answer because I don't want to just argue with you, but when I opened my Bible randomly to read this morning, it was this exact page my Bible opened to. So I will mention this.
Jonah preached to the people of Nineveh the message God told Jonah to preach and the people of Nineveh believed and repented of their sin.
If you read Jonah 2:3-6 They believed God's word, proclaimed a fast, and from the least to the greatest put on sackcloth.. Even the King put on sackcloth and sat in ashes.
These are the signs of mourning and repentance in the Old Testament..
Then they turned in repentance - in Jonah 2:8 we see even the animals were covered in sackcloth, and they turned from their evil ways and their violence.
It's kind of like saying the Taliban all came to faith in Jesus and repented if their sin, so God gave to them salvation. Jonah had a grudge against the people themselves and so he was angry about God saving them, but they did both believe and repent and have sorrow for their sins.. and God heard their prayer and saved them.
He didn't arbitrarily save these people.. Of course, they as a nation didn't stay in repentance and belief so some years later they ended up being destroyed anyway, but here we see repentance and sorrow for sin..
It's showing us that no one, not even the greatest enemies of God's people are beyond salvation, and no one should be angry when God saves Israel's enemies, but rather be happy and rejoice in God's providence...
This said, if your asking whether or not you can specifically ask a mixed group of believers or unbelievers to pray for you (and not walking up to a person of another faith and asking for prayer) I don't suppose it's particularly harmful. It wouldn't be a sin and your God is greater than he who is in the world..
My preference if I really wanted healing would be to not ask anyone except for a specific few people I knew were good Christian brothers and sisters, but I've also asked here for prayer in the prayer forum, and I know most who call themselves Christian aren't..
So it depends on your heart I would imagine, and whether or not it would bother you that people are praying to other gods in your name.
If you have no issue with it, then fine. If you want a healing that would absolutely glorify God, I would not ask, and I believe that to be the most Biblical position.
Although again. I don't believe in being rude to people either. Anyone who offers prayer I am always kind to..
For me it bothers, bothers to the point I have a more difficult time even asking nominal Christians for prayer.. but, that's me. I don't expect that every one would feel the same.
I pray for you to have a speedy recovery from your illness.