Keller refused to keep the court offices open after 5 p.m. on Sept. 25, 2007, when attorneys for Michael Richard said computer problems were delaying their efforts to file late appeals of his death sentence.
Richard was executed that night by lethal injection for the rape and murder of a Houston-area woman.
"It's one thing for a banker to close shop at five o'clock sharp. But a public official who stands between a human being and the death chamber must be held to a higher standard," Burnam said.
Keller, a Republican who has served on the court since 1994, did not immediately return a telephone message left late Monday at her office. Most state offices were closed for the Presidents' Day holiday. Keller's home phone number is unlisted.
Burnam's resolution calls on the House of Representatives to investigate Keller for gross neglect of duty. If the House finds grounds for impeachment, it would be up the state Senate to conduct an impeachment trial.
Republicans hold majorities in the House and Senate.
Texas has not impeached a state judge since the 1970s when a series of judicial scandals led to ethics reforms, Burnam's office said.
After the Richard case caused an uproar, the Court of Criminal Appeals decided to allow emergency e-mails in death penalty cases, rather than require attorneys to physically file the paperwork at the court.
The Texas court had been one of the few in the nation that did not accept filings electronically.
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