Where have I to deposit the fees?
may be called very old fashioned: 'have to' theoretically may be used in this way.
This swapping of positions of I and have is an example of what is called subject-operator inversion, or subject-auxiliary inversion. Auxiliary verbs are noted for their ability to take part in such tricks. But "have to" is said to be a 'semi-auxiliary': we may use it as an auxiliary or as a usual verb ("main verb").
In our time, people treat 'have to' as a main verb in questions, hence, no inversion is made. But in questions there must be an auxiliary verb in the first (operator) position. So we use another auxiliary verb - DO. This is called DO-support, and the proper sentence is:
Where do I have to deposit the fees?
Where do have I to deposit the fees?
is ungrammatical because it uses the S-A inversion and adds DO on top of that. It is as if we have decided to treat 'have to' as a full-fledged auxiliary, and then decided to throw in the DO-support for good measure. We end up with two auxiliaries in a slot where only one is allowed (either 'do' or 'have').
Here's an example of the old-fashioned use of "have to" as an auxiliary in a question, with S-A inversion:
But go on, why had you to go? (In a book by Iris Murdoch).
Notice how 'had' stands before 'you'. In modern English, we use DO-support in such questions with 'have to':
But go on, why did you have to go?