This is in a domestic garden, in Cornwall, UK. What is it?
Yes, they are tent caterpillars. No, you really don't need to burn them. Just rip the tent open and the birds should take care of the caterpillars, without their protective tent.
I know you've accepted an answer, but I'll add this, because, without knowing quite what they are, you may be endangering your or your family's health by trying to deal with them. Given where you are, these are likely to be Brown Tail Moth tent caterpillars - if you are asthmatic, or anyone in your family is, you have to be very careful not to inhale the fine hairs from the caterpillars, which may also be mixed in with the tent fibres. These hairs are also highly irritant to the skin, so protective clothing should be worn if you wish to tackle the tents yourselves, although this is best done in winter rather than now. When we had these in late spring three years ago in street trees here in West London, the area was cordoned off and a crew wearing full hazmat suits and breathing gear came to cut them out, paid for by the Council. Shouldn't think Councils can afford to do this any more, but it's worth a try - otherwise, advice from East Devon Council as to how to remove them safely in the link below