Which locations are traditionally Santa's home - i.e. where can you visit Santa Claus year-round?

North Pole is a town in Alaska whose main business is about Santa Claus.

You can also visit the tomb of his ancestor, Saint Nicholas, at Demre in Turkey.

In Ontario, a little south of what I would consider Northern Ontario, is a Santa's Village that is only open in the summer, though parts of it open briefly just before Christmas for SantaFest. Their tag line is

located half way between the equator and the North Pole, it’s where Santa spends the warm months relaxing and experiencing the joy of Christmas all summer long.

It's reasonably well-reviewed and is the top rated attraction in Bracebridge (a very tiny town in cottage country.)

The Canadian government has recently affirmed that Santa is Canadian and the North Pole is in Canada. Though the first claim was mostly to make another party look bad and the second was about shipping lanes and territory, but what the heck. The US got quite bent out of shape about it and the Germans are trying to get UNESCO to declare Santa to be German. So be aware that your geographical quest could turn political unless you're happy with many simultaneously true answers.

The Basilica of Saint Nicholas in Bari, Italy.

Saint Nicholas Church in Myra, Turkey.

Since Santa is a multi hybrid which all leads back to the Germanic god Odin. In pre-christian Europe it was the common believe that Odin was traveling with his flying eight-legged horse Sleipnir. Also in the pre-christian Europe the germanic people celebrated Yule on midwinter day. It was this celebration that was used to christianize Europe. A spar with lights so common in current Christmas tradition comes from this Yule festival.

(Source: Wikipedia)

In a christian europe, Saint Nicholas took the role of the roof walking saint throwing gifts through the chimneys. Saint Nicholas up until today is celibrated in various forms throughout Europe. In Austria Saint Nicholas is assisted by devil like figures called Krampus and in the Netherlands and Belgium Saint Nicholas is called "Sinterklaas" and has various helpers called "black petes". It is this figure that become the role model for Santa Claus. After the exchange of New York for Suriname between the English and the Dutch, Sinterklaas kept being celebrated and finally got transformed into Santa Claus.

Having this brief trip to history, to answer your question you have quite some options to visit Santa's house if you like. Ofcourse you can visit the ones already being mentioned, but since many of us avid travelers it is always nice to have multiple options. In the Netherlands the story is that sinterklaas comes from Spain. Why Sinterklaas comes from Spain remains a mystery. It might had to do with the spanish rule in the Netherlands. In their anthem, the Dutch are still loyal to the spanish king.

The most real figure is Saint Nicolas himself. He was the figure used to terminate the pegan worship of Odin. Visiting his "house" can be done in Myra, Turkey

Since, Santa is a fictional figure that still has its houses (see earlier houses), I would state that houses dedicated to Odin, which in the end is the "real" Santa Clause, I would consider the Rosala Viking Center a Santa Clause house as well.

There's Santa's Village in New Hampshire, USA.
It is a theme park which is mostly open during the summer months (but also on weekends throughout the fall and winter months).
There is a Santa Claus who lives there and greets guests, and the entire park is North Pole/Santa themed.

Very surprised that nobody has mentioned Lapland yet.

Here in the UK, Santa (or Father Christmas) is said to live in both the North Pole and in Lapland. The Noth Pole has already been covered by another answer though, so I'll give Lapland as my answer.

I don't know of any historical connections that would tie him to Lapland, so my guess is that this legend is probably largely due to the fact of reindeer being native to Lapland.

There's a fairly big tourist industry revolving around this, so I'm surprised nobody else has mentioned it yet.

Oh, and co-incidentally, Lapland is part of Finland... which brings us neatly back to the original question.

Here in the United Kingdom we look forward to a visit from Father Christmas who has become associated with the American Santa Claus. We always refer to him living in Lapland. This question is the first time that I've heard of him living in a very specific part of Lapland - Rovaniemi.

Since Christmas and Santa Claus is today a commercial event, the true contemporary home is where christmas is produced, not where it historically may have originated. so obviously, Santa has moved to China in the meantime. Specifically, to Xitan, where annually over 62 Mio USD worth of Christmas decoration are made.

There is a Santa Claus Lane in Carpinteria, CA, a little South of Santa Barbara. I don't know of a building to visit that is in the theme, however

Another option: Santa Claus, Indiana - they've got Christmas and holiday-themed stuff year round, as well as a theme park called Holiday World.