Replacing a fence usually does not require a variance. Assuming the residence is in a part of town zoned for residences, I would suspect that local ordinances allow for fences without the need for a variance. Replacing a fence may require application for a construction permit. I suspect there are facts not in your question which are missing; and which would, if provided, shed light on to your situation.
As a general matter, deed restrictions, including easements, must be recorded to have any restrictive effect on the deed. The easement for a future sidewalk must be recorded somewhere: either individually in the deed books in the country records, or in a master plan (which would also need to have been filed in the county records.) Also, keep in mind that the government almost always reserves the area closest to the street as a public right of way, on which the government can build a sidewalk.
I recommend you carefully review your deed language, and your title insurance. Your title insurance policy should mention all deed restrictions. Another possibility is to take the metes and bounds description as found in your deed and physically mark the boundaries on your property. You might be surprised to see the legal description (metes and bounds) does not extend your ownership of the land as far as you think it does.
At the very least, contact an attorney. If you had a hearing, of any kind, I hope you had an attorney represent you. If you did, bring this matter to that attorney's attention. If you did not, then immediately consult with an attorney.
Disclaimer: For a free consultation contact me directly at 973-519-0196 or adam@LefkowitzLawOffice.com. If this answer was helpful, please mark it as helpful or as a best answer. Answering questions on avvo.com does not create an attorney-client relationship. You are strongly encouraged to seek and retain your own counsel. Information and answers posted to this website are for general informational purposes only.
It seems like your question clearly lays out the disadvantages of having a restriction like this. As for what you can do, you will need to consult with a local attorney. In some limited situations municipal easements expire over-time, but most are perpetual and do not expire. An attorney will need to review the easement, and apply New Jersey law to the facts to determine what your rights are. You may have to consider putting the fence on your own property that is unencumbered by the easement.