A policy you can attach to a role to enable the caller to read from Athena tables. s3:PutObject is required to put the query results CSV in the query result bucket.

IAM Role AttachedPolicy for reading from Athena Tables.json

{
    "Statement": [
        {
            "Action": [
                "s3:ListBucket",
                "s3:GetBucketLocation"
            ],
            "Effect": "Allow",
            "Resource": [
                "arn:aws:s3:::bucket-your-data-is-in",
                "arn:aws:s3:::bucket-specified-for-athena-query-results",
            ],
            "Sid": "FindAndScanStagingBuckets"
        },
        {
            "Action": [
                "s3:GetObject",
                "s3:PutObject"
            ],
            "Effect": "Allow",
            "Resource": [
                "arn:aws:s3:::bucket-your-data-is-in/some-subfolder/*",
                "arn:aws:s3:::bucket-specified-for-athena-query-results/*",
            ],
            "Sid": "GetAndPutAthenaAndPublicResults3Objects"
        },
        {
            "Action": [
                "glue:GetTable",
                "glue:GetTables",
                "glue:SearchTables",
                "glue:GetTable",
                "glue:GetDatabase",
                "glue:GetPartitions"
            ],
            "Effect": "Allow",
            "Resource": [
                "arn:aws:glue:us-west-1:YOURAWSACCOUNTID:*"
            ],
            "Sid": "AccessGlueParts"
        },
        {
            "Action": [
              "athena:StartQueryExecution",
              "athena:CancelQueryExecution",
              "athena:StopQueryExecution",
              "athena:GetQueryExecution",
              "athena:GetQueryResults",
              "athena:GetWorkGroup"
            ],
            "Effect": "Allow",
            "Resource": [
                "arn:aws:athena:us-west-1:YOURAWSACCOUNTID:*"
            ],
            "Sid": "EnableAthenaQueries"
        }           
    ],
    "Version": "2012-10-17"
}