I own a tavern on a different side of the land I'm on and I have things from it to make sure I go back with no flaws. There is a war raging across all the land so it appears to have been destroyed. Would I no longer be able to teleport even to where the rubble remains?
The rules for what counts as an "associated object" aren't particularly specific, so it'll be up to your DM to decide if whatever you took from your tavern counts as an associated object for the ruins of your tavern. If they decide it does, then you should be able to teleport there with perfect accuracy. If they decide it doesn't, then you'll be teleporting to a location that no longer exists. Teleport includes specific rules for this circumstance.
“False destination” is a place that doesn’t exist. Perhaps you tried to scry an enemy’s sanctum but instead viewed an illusion, or you are attempting to teleport to a familiar location that no longer exists.
If you consult the table provided in the spell description, you can see that, when teleporting to a false destination, you have a 50% chance of teleporting to a similar area, and 50% chance of suffering a mishap. Since a mishap causes a reroll on the table, you will inevitably end up teleporting to a similar area. What the dice will determine is how much damage you take getting there.
Similar Area. You and your group (or the target object) wind up in a different area that’s visually or thematically similar to the target area. If you are heading for your home laboratory, for example, you might wind up in another wizard’s laboratory or in an alchemical supply shop that has many of the same tools and implements as your laboratory. Generally, you appear in the closest similar place, but since the spell has no range limit, you could conceivably wind up anywhere on the plane.
So if you try to teleport to your tavern, you'll end up at the closest tavern, possibly bleeding out from force damage you took along the way.
However, you know that it's been destroyed, so you're not going to try to teleport to the tavern - you're going to try to teleport to the rubble where it used to be. That gives you some more options.
If you can find a way to take a look at the ruins (perhaps you already have, given that you know that it was ruined; if not, Scrying is recommended), you can teleport to it as a "viewed once" destination. If not, you can try as a "description" destination.
“Viewed once” is a place you have seen once, possibly using magic. “Description” is a place whose location and appearance you know through someone else’s description, perhaps from a map.
In either case, your accuracy is greatly improved over "false destination", but still leaves a lot to be desired. You'll have a 27% chance of arriving at your intended destination, 20% chance of arriving some number of miles away, 10% chance of arriving at a similar area (although that wouldn't be too bad - the closest ruin to your tavern is likely to be nearby), and a 43% chance of a mishap.
I don't see why the associated item to the tavern would not be associated with the damaged tavern. It is subject to DM ruling though.
Assuming the surrounding landscape has not been completely torn apart, you are probably very familiar with the outskirts of your tavern, probably the street right in front of it.
Even so, the general geographical location of a place you've lived in (and the tavern is probably in your home town) is very familiar too. Say your tavern is located 200' north-northeast from the Worthington bridge in Calimbury. Just concentrate on that geographical location.
Now if Calimbury has been cut off from the prime material and swapped by a hellish landscape outta Carceri, all bets are off. And that is probably not the case.