Can PostGIS use spatial indexes on foreign tables?
Assuming current Postgres 9.6, it can work.

Postgres can use indexes on a remote Postgres server. You may have to do more than for local tables, autovacuum does not run ANALYZE for remote tables automatically. Start by reading the chapter "Remote Query Optimization" in the manual.

Related answer with more details:


Postgres 9.5 foreign table inheritance not using indexes


Since PostGIS is an extension, this new feature of Postgres 9.6 is particularly relevant:


  Allow extension-provided operators and functions to be sent for remote
  execution, if the extension is whitelisted in the foreign server's
  options (Paul Ramsey)
  
  Users can enable this feature when the extension is known to exist in
  a compatible version in the remote database. It allows more efficient
  execution of queries involving extension operators.


Bold emphasis mine.

The use of indexes is bound to operators. Various PostGIS spatial functions are transformed to use operators internally when the query is planned. This related answer on SO has more explanation:


Can PostgreSQL index array columns?


Assuming Postgres 9.6 for local and remote server, and the same version of PostGIS installed, running this on the local server should do the trick:

ALTER SERVER your_foreign_server_name OPTIONS (ADD extensions 'postgis');


Here is a related blog post by Michael Paquier.