"No new inning" means that no new inning may start once time expires, but that the current inning can be completed.
"Drop dead" means that the game ends no matter what. In such a case, the whichever team was ahead in the previously completed inning is the winner.
Managing the end of the game properly can mean the difference between a win and a loss in a close game. It can lead to some unconventional actions such as intentionally making outs or intentionally not swinging at strikes to reach or delay the time expiration.
Update: After letting this article marinate in my brain for a day or two after it was written, I decided that the information would be best presented via table instead of all text as before, so I updated the article to include the above table and eliminated a lot of the verbiage in the article.