SQL Server Execution Plans and Index Drops
Welcome to DBA.StackExchange, and interesting question!...one I've never thought of before. I had to double check myself on what I thought the process was, so I think this StackOverflow Answer should be what you're looking for. Specifically the first statement of the answer:

Dropping/Rebuilding an Index will result in invalidation of any cached
execution plans using this table/Indexed view. And sql server will
generate a new execution plan on next execution.

Essentially, the plans in the plan cache will automatically be marked as invalid of any existing execution plans that reference the dropped index. Then the next time a query runs that involves the entity of which the index was dropped from, a new execution plan is automatically generated and cached in the plan cache because the old one is invalidated. (Eventually the invalidated plans are flushed from the plan cache to make room for new plans being cached.)
Here's an additional resource that concurs this is how it occurs: Dropping unnecessary indexes - effect on query plans in SQL Server 2005