I take down gorse regularly as part of a national trust group, all trogged up in thick, long leather gloves, and even then, with a few of you going at it, a few pushing/pulling and another cutting, it's still very painful.
Do be careful burning it, it's remarkably easy to get a gorse bonfire very wrong, the needles burn off very quickly with copious amounts of smoke, but the wood remains, and takes a lot more to get it going. Therefore if you're not careful, you end up with a pile of blackened stalks that isn't likely to go up and no heat at the bottom - a so called hollow fire. To minimise the problems, you want a decent stock of old, dense and dry wood (fence posts are good) to pile on periodically to weigh the tangled mess down. Also make sure you've got a decent fire burning before you try to load any gorse onto it, wood planings are ideal, with plenty of smaller twigs and then progressively larger wood, not gorse. If you've got any ash, then getting that into the fire early on will make your life much easier. Also try to lay all the gorse on in one direction, good technique for any bonfire but notoriously difficult with gorse.
Good luck, get a good pair of gloves and try to avoid getting spiked too much. If you're using a chainsaw, make sure you wear all the protective kit - helmet and trousers at the very least.