Once you gain access to a new tier, you can access it on any gate at any time. However, there is a crown fee to do so if you haven't cleared the previous tier at least once on that gate, outlined in Metaphysic's post.

You can clear the crown fee for skipping to T2/T3 on a gate by clearing the last stage of the previous tier. This will last until the gate cycles out. So if you intended to run T3 a lot in the Jade Titan Gate, for example, you might want to clear Jade Titan's T2 at least once first, to clear the 500 crown fee.

If, some time later, another Jade Titan Gate cycles in, it won't "remember" that you cleared T2 on an earlier Jade Titan Gate. So you'll have to re-earn any free clearances.

Free clearance lasts only until the gate cycles out. This gives you an incentive to occasionally run a lower tier on a new gate-- at least you'll remove the crown fee for running the tiers you REALLY want to run in that gate.

Oh, and, the fees for Tier 2 are independent of the fees for Tier 3. You do not have to clear the fee for Tier 2 in order to clear the fee for Tier 3. You can skip to Tier 2 and pay the fee, then clear your way to Emberlight, and then you'll have free access to Tier 3 but not Tier 2. But if you only wanted to remove the fee from Tier 3, you probably don't give a hoot about removing Tier 2's fee.

You can see your free access status on the Progress tab of your Character screen. Scroll down to the chart with the gate icons. The starred tiers are the ones you can access for no crown fee.