How to assign selected values from a table to specific variables in Pl/pgSQL?

For example, here is what I am trying to accomplish:

EXECUTE 'SELECT name, address INTO variable1, variable2
         FROM employee WHERE id=1';

Better alternative

Actually, there is nothing indicating a need for dynamic SQL. All identifiers are stable. Assuming you only want to parameterize the id value, simplify to:

SELECT name, address FROM employee WHERE id = $1
INTO   variable1, variable2

Or:

SELECT INTO variable1, variable2
            e.name,    e.address
FROM   employee e
WHERE  e.id = _my_variable;

The according chapter in the manual: "Executing a Query with a Single-row Result"

Dynamic SQL would be indicated when working with parameterized table or column names.
Here is a list of related question for dynamic SQL

Cause for immediate problem

The reason why your original code did not work, per documentation:

SELECT INTO is not currently supported within EXECUTE; instead, execute a plain SELECT command and specify INTO as part of the EXECUTE itself.

a_horse demonstrates valid syntax in his answer.

Either alternative relies on the query to return a single row. If your id is defined unique, that cannot break.