Cocaine should not burn. Yes, it must be cut with something although it's hard to say exactly what it's cut with. If it was just one person, you could say maybe they have a very sensitive nose (although even this is unlikely, IMO, given cocaine's anaesthetic properties) but as you say it's a group of you, that's especially unlikely.
You might try an acetone wash, you can find instructions maybe here, apologies I forget the thread if there is one and can't be bothered to look for it, however I just did a quick
Google (nope, goddammit)
DuckDuckGo for you and there are a lot of pages describing this process.
That said... while it would remove a lot of inactive cuts, there are some active cuts that would survive this process, I am pretty sure. What are the subjective effects like? Does it feel like cocaine? There are a whole bunch of active cuts I could think of that would burn a lot when insufflated... but, subjective feelings are such an unreliable measure of substance purity, it's just so hard to tell and the power of suggestion is so strong.
Either way - coke should not burn, I would bring it up with your source if you have that kind of relationship, or if you don't, just try to source from elsewhere in future.