I'm not sure I understand. Foreclosure means the lien holder takes title to the property, and the current owner, the one who owes the debt that the lien holder was unable to collect, loses title. Then the lien holder, who now has title to the property, sells it for whatever the market will bear, and pockets the sale price minus whatever costs he has to pay. That effectively means the lien holder gets whatever equity is in the property once he has foreclosed on it and sold it.
This article talks about DC tax lien sales, and doesn't mention keeping equity
Well, of course not, since it only talks about buying the tax lien and foreclosing on it; it doesn't talk about the part where once you've foreclosed, you sell the property and pocket the money.
The article does describe the foreclosure process the same way I did above:
"The District of Columbia statutes require that you file a judicial foreclosure on your lien. This means that your attorney will run title, notice all the parties on title and file a foreclosure action in the court to get a judge to issue a final judgment. Once the judge rules in your favor, you can go to the DC Office of Tax and Revenue, pay any outstanding taxes and bills that may be due, and get issued the tax deed."
So you buy the tax lien, and if the tax isn't paid, you foreclose, you pay off the outstanding taxes and bills, and you get the property. The now former property owner, whom you've just evicted, gets nothing. Then you can sell the property for whatever the market will bear. So who has the equity?
Not normally, no. Usually the lien holder gets the value of the lien and the remainder of the sale proceeds go to the owner. If the owner is really "left with nothing" he didn't actually have any equity in the house.
In some rare cases, with underwater mortgages, the lender may be allowed to take title.
I do see plenty of web references that describe foreclosures the way you do, but they all talk specifically about foreclosures when the lien is a mortgage. Does the same process apply when the lien is a tax lien? What information I can find suggests it doesn't--or at least, there seems to be a lot of wiggle room.