well it depends on the fungus really. as the tree is dead the fungus over time will weaken the structure and one day [who knows when] it will collapse. the advice is for safety really.

Perhaps when the rose has finished think about it then.

do you know which fungus?

Beware if you have other trees nearby as the fungus spores will be transported via the wind and could spread, though I don't think that would do much harm to healthy trees, the main thing to be concerned about has already been suggested to you by Seaburngirl, and that is the chance that the dead tree will now be in danger of collapse once the fungus has eaten it, but again it depends just how far developed the fungus is, it can take years

Have a look at some pictures of honey fungus. If its that you need to remove it asap.

Thanks so much all for really useful comments. I don't 'think' it's honey fungus, but I'll attach a photo. The fungus is right at the top of the branch stumps. However what with this and a large ants nest at the base, we're starting to realise that the tree's days are numbered, as the bark is coming away from the trunk.

However we'd like to wait until the autumn if not the winter before we chop the tree down due to the rambling rose. I'm hoping the rambling Rose can be untied because it hasn't actually 'attached' itself to the tree thankfully.