The following command will export a list of phpBB email addresses and put each email address on a new line (so you can copy in paste into Excel or another app).

Export phpBB emails from MySQL

The following command will export phpBB email addresses from the phpbb_users table:

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mysql YourphpBB_db -u root -p -BNe 'select user_email from phpbb_users' > /tmp/outfile.txt

I know there are other option to export email addresses from phpBB but I find this the easiest / fastest way (I exported about 37,000 email addresses in a second or two on a reasonably low spec CentOS 6 VPS.