If a tax lien isn't paid by the current owner of a property and the owner of the tax lien certificate files for foreclosure, what happens next? 

It's my understanding that the property would then go to the courts to issue the tax deed foreclosure and then from there what happens? Does the owner of the tax lien certificate then have to bid on the foreclosed property? I've been trying to do research, but somethings contradict others.

I have watched some people explain it a few different ways and the simple explanation was you buy a tax lien, if it's unpaid, the tax lien cert. holder can file for forclosure and then they get the property. But that doesn't seem right?

Can someone explain the process to me?

I'm in New Hampshire if that matters. 

You may want to talk to your local taxing authority.  I don't believe tax lien certificates are sold in New Hampshire.  Two years after a lien is placed on the property, the town can simply take it and sell it, but they don't convey the right to take it and sell it to anyone else (which is what a lien certificate represents). However, there is a 3-year buy-back period, which complicates things in NH. I think most towns put the deeded properties up for auction, but I have seen some of them listed in the MLS with a local broker.

In NH, I believe it varies from town to town exactly how things are handled. I purchased a vacant lot in a NH town at a tax auction in 2019.

In that town the clock starts when they send out the first tax bill for the current year (in this case May). If the owner has not paid taxes by  the following years tax cycle (again May in this case), then they give the owner 2 years and a day to pay the back taxes (all the while assessing penalties and fees). If after this period is up, there has been no attempt to pay, and no communication, then the town will simply take the property. 

In that particular town, they usually wait till they have at least a couple of properties and then do a public auction (once a year, always in September). 

My understanding is that the former owner has 3 years from the date the Town records the quit-claim deed to come back to the town and reclaim the property (by paying all back taxes, fees, penalties etc). After that 3 year period the former owner looses all rights to the property. I'm not sure what happens if the property was sold at auction before that 3 year period ends, and the owner comes back to take the property?

It's also worth noting that there is a 10 year "quite" period from the date the quit-claim deed is recorded, in which you cannot re-sell the property with a warranty deed (but you can resell with another quit-claim deed).

This particular town also does not sell tax liens to the public. They try their best to work with the owner, but if that fails, then they only ever take ownership of the property to sell at public auction.

This doesn't exactly answer you questions, but hopefully it provides a bit of insight.