When you e-file, you should expect 3 emails. 1 confirming transmission, 1 that confirms acceptance or rejection by the IRS, and a 3rd that confirms acceptance or rejection by your state (or DC in this case). Sometimes those emails are caught in a SPAM filter, that's up to you to make sure you receive them.
If you e-file your federal return and don't e-file your state return using Turbotax online you will often get an email reminder, this can also go to spam. If you filed using the desktop version, you don't get reminder emails, your filing is all under your control and your responsibility.
At any time after e-filing, if you log into your online account (if filing online) or if you open your tax file (if you use the desktop version) you will get an update on your filing status. It's also certainly your responsibility to watch your bank account and notice if your tax due was not withdrawn or your refund not paid to you.
