Two suggestions for you since a relative newbie in SQL. Use "aliases" for your tables to help reduce SuperLongTableNameReferencesForColumns, and always qualify the column names in a query. It can help your life go easier, and anyone AFTER you to better know which columns come from what table, especially if same column name in different tables. Prevents ambiguity in the query. Your left join, I think, from the sample, may be ambigous, but confirm the join of B.ID to TB.ID? Typically a "Team_ID" would appear once in a teams table, and each blabbing entry could have the "Team_ID" that such posting was from, in addition to its OWN "ID" for the blabbing table's unique key indicator.
SELECT
B.id,
B.mem_id,
B.the_blab,
B.blab_date,
B.blab_type,
B.device,
B.fromid,
TB.team_id
FROM
blabbing B
LEFT JOIN team_blabbing TB
ON B.ID = TB.ID
WHERE
TB.Team_ID IN ( you can't do a direct $team_array here )
OR B.mem_id = SomeParameter
OR b.FromID = AnotherParameter
ORDER BY
B.blab_date DESC
LIMIT 20
Where you were trying the $team_array, you would have to build out the full list as expected, such as
TB.Team_ID IN ( 1, 4, 18, 23, 58 )
Also, not logical "||" or, but SQL "OR"
EDIT -- per your comment
This could be done in a variety of ways, such as dynamic SQL building and executing, calling multiple times, once for each ID and merging the results, or additionally, by doing a join to yet another temp table that gets cleaned out say... daily.
If you have another table such as "TeamJoins", and it has say... 3 columns: a date, a sessionid and team_id, you could daily purge anything from a day old of queries, and/or keep clearing each time a new query by the same session ID (as it appears coming from PHP). Have two indexes, one on the date (to simplify any daily purging), and second on (sessionID, team_id) for the join.
Then, loop through to do inserts into the "TempJoins" table with the simple elements identified.
THEN, instead of a hard-coded list IN, you could change that part to
...
FROM
blabbing B
LEFT JOIN team_blabbing TB
ON B.ID = TB.ID
LEFT JOIN TeamJoins TJ
on TB.Team_ID = TJ.Team_ID
WHERE
TB.Team_ID IN NOT NULL
OR B.mem_id ... rest of query