Last July, Sheehy's wife, Connie Sheehy, filed for divorce after nearly 29 years of marriage. Her divorce filing stated the "marriage of the parties is irretrievably broken."
Heineman declined to answer questions about what had occurred that led Sheehy to resign. He also would not say whether he asked Sheehy to resign or if Sheehy acted on his own.
Heineman said he will begin the process Monday of looking for a new lieutenant governor.
If something were to happen to Heineman before he selects a new lieutenant governor, Republican Speaker of the Legislature Greg Adams would ascend to the governor's post.
Sheehy's announcement shakes up an already turbulent 2014 governor's race. Another Republican candidate, former Nebraska Speaker of the Legislature Mike Flood, entered the race briefly but withdrew in December after his wife was diagnosed with breast cancer. State Sen. Charlie Janssen of Fremont, a Republican, has also said he is leaning toward running.
Several Democrats have signaled an interest in running, including University of Nebraska Regent Chuck Hassebrook and Nebraska state Sen. Steve Lathrop of Omaha. In recent weeks, party activists have also approached state Sen. Annette Dubas, of Fullerton.
"We're going to have a very strong candidate in 2014," said Vince Powers, chairman of the Nebraska Democratic Party. "This doesn't change anything, other than it really demonstrates that when you have one party in power for too long, arrogance and corruption and scandal follow it. It doesn't matter if it's Democrats in power or Republicans in power."
In an email to The Associated Press, Hassebrook said he is "taking a hard look" at running for governor in 2014.
"I recognize that no human is without failing. So I wish Rick Sheehy and his family the best as they seek to heal from this disappointing episode," he said.
Sheehy changed his Twitter account settings to private before Heineman's press conference. Shortly after, his account appeared to have been deleted.
Associated Press writer Margery Beck contributed from Omaha, Neb.