The main issue is that they can go off when they are not supposed to, which may cause more of a hazard than wearing no jacket at all.
Older/cheaper styles were/are activated by a dissolving tablet which, over time, would degrade due to getting damp and can then activate themselves at random. Whilst this can be no more than an embarrassment in some situations (and mildly entertaining in others!), it can be a real hazard in restricting movement or preventing access to other safety gear, release devices etc.
Newer/more expensive ones are activated by hydrostatic pressure (ie a head of water) which is an improvement. However, even this can still lead to problems when they activate due to the same issues of restricted movement etc.
Personally, I can think of nothing worse than being restrained by a climbing line, potentially to a point some distance above you, and restricted in my ability to move in potentially fast flowing water.
Despite having more than 90 auto-inflate lifejackets for our various boats (we charter out boats in the summer months and they are specified by the MCA) we still use fixed buoyancy jackets when working in or close to rivers (we maintain approx 140 sites for the EA each winter). Similarly, most of the diving/workboat contractors I work with also use fixed buoyancy jackets for their deck crew. Even the manual gas inflate jackets can be a hazard if the toggles get snagged and inflate the jacket (the last time it happened to me I was trying to get out of a locker - very amusing for those who were watching:sneaky2:)
IMHO, given that most arborists will not be working alone then if your RA advises that some kind of buoyancy aid is required then I would look at fixed rather than inflatable types. These have the further advantage that they can be worn under a chest harness etc if it so suits the individual. Playing devil's advocate, there could also be a case for NOT wearing a jacket if working in a tree but secured by suitable climbing lines etc.