There is always something; it is just what you are willing to do.
Selecting fewer columns can improve performance, removing functions can improve performance, reducing the result set can improve performance.
Fewer columns may require a different index to get the best performance, but indexes are a tricky beast when doing performance tuning. Each index you add will slow down INSERT, UPDATE and DELETE performance.
Removing the function may help performance too as it is presumably a calculation being done on Column3 or possibly a lookup related to Column3 or it could just be returning Column3 after doing no real work. I cannot see that function, so it is hard to say if removing that will help or not. Depending on what Function1 does, you might be able rewrite the query without the INNER JOIN. You may get a performance boost by changing the INNER JOIN to a WHERE clause since you aren't actually using any columns in that subquery apart for filtering the result set.
Since you are not filtering the result set, but looking at every row, even if you do switch the SCAN to a SEEK, you are still going to be essentially doing a SCAN since you are looking at every row.
Reducing the result set (ie adding a WHERE clause and filtering out some of the data) will likely improve performance, but likely isn't what you are wanting to do. Heck, adding "WHERE 1=0" will improve performance but will give you no results.
Since your subquery has no WHERE clause, you are going to be looking at all rows in the table. The main query is doing a SEEK, but looking at every rows, so it's going to be essentially a SCAN. Since the query will need to look at all rows with every execution, as the table grows in size, it will get slower.
If you are wanting to remove the INNER JOIN, I THINK this query should be identical and should have similar performance (test it though):
set statistics io on;
set statistics time on;
FROM Schema1.Object2 Object1
WHERE Object1.Column3 = Function1(Object1.Column3)
My understanding, the above query should be identical to what you had but only 1 lookup on the table. It'll still be a SCAN because we are needing to look at each and every row, but only a single table lookup SHOULD be faster than 2 table lookups.