We definitely wouldn't recommend it for the following safety reasons, whether in a stove or an open fire. Green, unseasoned wood, which has a high moisture content and in particular the excessive amount of sap (sticky resin) contained in the different varieties of Christmas trees, can cause serious problems inside your flue or chimney. This is because flue gases should be hot enough to rise and safely carry away the gas, including carbon monoxide, and any particulates (soot) that it contains.
The moisture contained in the unseasoned wood will significantly lower the flue gas temperature so that creosote is formed and any unburned resin, mixed with soot, will stick to the inside of the chimney potentially resulting in a chimney fire. Apart from that you'll cause nuisance smoke which is illegal (even if you're using a Defra smoke exempt appliance) and also end up with unsightly and dirty, difficult to clean, stove glass. Even if you fully seasoned the wood from your Christmas tree it really isn't worth the effort. Instead, use your local authority's environmentally friendly Christmas tree disposal service.