How can I get a list of all columns referenced in a stored procedure?

When a stored procedure is executed it is parsed and compiled into a query plan, this is cached and you can access it via sys.dm_exec_cached_plans and sys.dm_exec_query_plan in XML format. The query plan records the 'output list' of each section of the parsed code. Seeing which columns are used by the stored procedure is just a matter of querying this XML, like this:

--Execute the stored procedure to put its query plan in the cache
exec sys.sp_columns ''

DECLARE @TargetObject nvarchar(100) = 'sys.sp_columns';

WITH XMLNAMESPACES (
    'http://schemas.microsoft.com/sqlserver/2004/07/showplan' as ns1
), CompiledPlan AS (
    SELECT 
        (SELECT query_plan FROM sys.dm_exec_query_plan(cp.plan_handle)) qp,
        (SELECT ObjectID FROM sys.dm_exec_sql_text(cp.plan_handle)) ob
    FROM sys.dm_exec_cached_plans cp
    WHERE objtype = 'Proc'
), ColumnReferences AS (
    SELECT DISTINCT
        ob,
        p.query('.').value('./ns1:ColumnReference[1]/@Database', 'sysname') AS [Database],
        p.query('.').value('./ns1:ColumnReference[1]/@Schema', 'sysname') AS [Schema],
        p.query('.').value('./ns1:ColumnReference[1]/@Table', 'sysname') AS [Table],
        p.query('.').value('./ns1:ColumnReference[1]/@Column', 'sysname') AS [Column]
    FROM CompiledPlan
        CROSS APPLY qp.nodes('//ns1:ColumnReference') t(p)
)

SELECT 
    [Database], 
    [Schema], 
    [Table], 
    [Column]
FROM ColumnReferences 
WHERE 
    [Database] IS NOT NULL AND 
    ob = OBJECT_ID(@TargetObject, 'P')

Caveat emptor this depends on how you define 'used'. It may be that a CTE within your stored procedure references 5 columns from a table, but then when this CTE is used only three of the columns are passed on. The query optimizer may ignore these extra fields and not include them in the plan. On the flip side the optimizer may decide that it can make a more efficient query by including extra fields in an output to enable it to use a better index later on. This code will return the columns used by the query plan, they may not exactly be the columns that are in the stored procedure code.