Can sick leave make burnout worse?

The short answer to this is yes.

But some explanation is worth sharing

Sick leave does have a place but.....as part of a sound, well structured approach to taking burnout seriously and realising that it needs reversing. Merely getting allocated repeated episodes of sick leave or sequential sick leave ( when the burnout is not mending)  are not generally helpful and may even help to embed burnout further or at times can make the burnout worse and more difficult to shift.

So the wholesale use of sick leave in burnout management needs to be seriously questioned.

What is needed is a plan - in which sick leave may or may not need to play a part. 
We call this "strategic burnout management" and it requires that all aspects of burnout are evaluated.

The longer burnout has been lurking and the more severe it is - the more likely it is that someone may need some sick leave.

BUT - Sick leave is NOT a treatment for nor a cure for burnout