Does a Boggart react to a Dementor? If so, would it turn into a Patronus?
A Boggart likely wouldn't react to a Dementor
Dementors can't see
The trigger for a Boggart's transformation isn't explicitly stated, but it appears to be based on sight. Boggarts have an unknown form when not being observed, but this has never been seen by anybody.
So the boggart sitting in the darkness within has not yet assumed a form. He does not yet know what will frighten the person on the other side of the door. Nobody knows what a boggart looks like when he is alone, but when I let him out, he will immediately become whatever each of us most fears.
Harry Potter and the Prisoner of Azkaban
If the transformation is based on the Boggart being seen, it's impossible for them to transform when encountering a Dementor, as Dementors are blind.
Dementors may not fear anything
A Boggart's primary "weapon" is fear; it transforms into a representation of whatever its victim most fears. Dementors, as beings who cause and feed upon fear and despair, may not be capable of experiencing fear themselves. When Harry confesses to Lupin that he was thinking of a Dementor, rather than Voldemort, when the class was first introduced to one, Lupin says that it shows that what Harry fears most is "fear itself". If Dementors don't fear anything, there's nothing for a Boggart to transform into.
Boggarts may not react to other magical creatures in general
For the sake of completeness, it's possible - though I'm not aware of any explicit evidence to support it - that Boggarts only target (part-)humans, and therefore wouldn't react to Dementors (or other magical creatures).
Boggarts likely only affect humans, not creatures or non-beings.
Boggarts seem to only affect humans. They’re described as turning into the worst fear of the person who encounters them, which implies their ability is specific to people. At least some creatures (magical and mundane) likely do have fears, such as predators, but there’s nothing mentioned about Boggarts turning into a Crup or a Jarvey around a gnome.
A Boggart is a shape-shifting creature that will assume the form of whatever most frightens the person who encounters it.
- Boggart (Pottermore)
Boggarts are non-beings that seem to feed off of human emotions. Since it seems to be human emotions that sustain them, it’s likely their ability is specifically to affect humans, not necessarily any other creatures. In addition, boggarts and Dementors are both non-beings, and it’s unclear if they have any emotions that other non-beings could potentially sustain themselves with.
It is one of the strange non-beings that populate the magical world, for which there is no equivalent in the Muggle realm. Boggarts can be made to disappear, but more Boggarts will inevitably arise to take their place. Like poltergeists and the more sinister Dementors, they seem to be generated and sustained by human emotions.
- Boggart (Pottermore)
It seems very unlikely that Boggarts would have an effect on Dementors, since Dementors are non-beings, and as such likely don’t have any emotions or fears of their own to be fed on - both Boggarts and Dementors sustain themselves on the emotions of humans.