How could Harold Finch keep his fortune while faking his own death?

As Vedran Šego and Oliver_C mentioned in the comments, a lot of his assets are held by aliases. For example:

Throughout the show, it's heavily hinted that Harold Finch is also an alias, and it's not the name of the man who died, that was Harold Martin. Harold Finch is very much alive and the owner of IFT, a software company. It's not really mentioned how big the company is, but as they get government contracts (the machine) and have a huge New York City headquarters, it's presumably worth a lot too, though it's hinted in the pilot:

Finch: You see, Mr. Reese, until that day [9/11/01], I had spent the better part of my life making myself very rich. Suddenly all that money didn't seem to amount to much.

In episode 6- The Fix, as Harold Partridge (Investor), he invests in Virtanen Pharmaceuticals while investigating them, and at the end of the episode, Shorts the company just before they assist in having the owner arrested, which makes him a considerable amount of money:

Finch: I know you don't care who you hurt to get what you want. I know the only thing you do care about is money. So that's what I'm going to take from you. Your money. All of it. Your were right Mr Keller. I never will have to invest in another company

Beginning in episode 12, Legacy, we see Harold Wren who is an insurance underwriter for Universal Heritage Insurance, this is a well established identity tied to the name he graduated MIT with. The job title combined with his very fancy office seems to indicate this is a wealthy alias.

Additionally, his aliases have given him inside knowledge into locations which are abandoned indefinitely, such as The Library as he mentions in the pilot:

Finch: This building was sold to a bank that I control, which promptly declared bankruptcy so the property's in a kind of limbo, it doesn't exist

Not having to pay for property saves a lot of money in the long run.