Empty describes something that contains nothing. So, for example, it can refer to a container or a room that has nothing inside; it can refer to a seat that has nobody sitting on it, or for example a threat which lacks sincerity of intention (or means to carry out).

Vacant, simply means unoccupied; so, just like "empty", it can be used to describe a seat that is available, however unlike empty, you would not use the word "vacant" to describe a container that has nothing in it (you could, but it would sound strange).

Unfilled, means that the maximum capacity of something has not been utilised. Again, in the case of the seat, it is an unfilled seat.

It can also have a meaning as that of "vacant", for instance you might have an unfilled position in your work - that means you have a position (e.g. manager role) that needs to be taken up by somebody.

However, something need not be empty to be unfilled; for example, think of a train that has spare capacity, but is not totally empty - it is therefore unfilled.

Unfilled is commonly used to describe a job vacancy that either nobody has applied for, or the candidates are not suitable or possibly all have been rejected. Therefore the company/person/organisation, continues to advertise the job vacancy, therefore there is still an unfilled vacancy.

Empty is in general conversation used to describe a container or a space that contains nothing. For example "My food cupboard is currently empty, because I always eat everything in one week and then restock for the next week."

Vacant can mean empty but it also has the meaning of unoccupied. 

The seat situation is perhaps not the best example to understand the full nuances. They can all be interchanged and an overlap occurs.

To use public transport as an example.

"please leave this set vacant for disabled people or pregnant mothers and people with children"

"Please move along and leave the end seats empty for other people to come in"

"please fill the front seats first so that late arrivals can use the rear seats without causing any disturbance"

I have never known or heard nor seen a sign saying "UNfilled seats" 

I have seen and heard UNfilled buses, stadiums, theatres etc, but it is very unusual and a little rare.

Fill and Unfilled refers to putting as much as possible into the available space or container, it generally does not apply to each singular seat. You can have unfilled trucks/jam jars/warehouses/buses/aircraft etc but not individual seats, at least it is not common to the best of my knowledge in BR English. To use that word like that.