No. Skipping a Merchant never removes a relic from your run. Entering one does: its three relics leave the relic lists whether or not you buy them.
How relic lists work
At the start of a run, Slay the Spire shuffles each relic rarity (common, uncommon, rare, shop, boss) into its own list. Relics you have not unlocked are left out.
- Elite rewards, chests and other rewards take relics from the front of a list.
- The Merchant takes relics from the back of a list.
- Every relic taken is removed from its list for the rest of the run.
What entering a shop does
When you walk into a Merchant, it stocks 3 relics from the back of the lists: two of a rolled rarity (48% common, 34% uncommon, 18% rare) and one Shop relic. Those relics are removed at that moment. Relics you leave on the shelf do not go back.
With The Courier, each relic you buy is restocked from the back of the lists as well.
Why skipping changes nothing for rewards
Rewards come from the front of the lists and shops from the back, so a shop you skip does not change which relic your next elite or chest gives. The shop's rarity roll also uses its own random stream, separate from the one that rolls reward relic rarity.
The two ends only meet when a list runs low. When a rarity's list is empty, rewards move to the next rarity up, and an empty rare list gives Circlet.
See relic pools for prices and odds and the shop and removal calculator.