Video footage showed Lycett shredding the cash at noon on Sunday, as per his pledge.
The comedian, wearing a flamboyant coat and ear defenders, said nothing as he was seen putting bundles of cash into a large red shredder, before curtseying to the camera once the the task was complete.
He shared the footage on Twitter with the caption: "A platform for progress".
Lycett had challenged Beckham to withdraw from his deal with Qatar before the deadline, saying the move would be symbolic of the former footballer's reputation as a gay icon.
Beckham has been criticised by many, including Lycett, for his £10m deal, due to the country's stance on gay rights with homosexuality illegal in Qatar.
In a video posted on Twitter, Lycett applauded Beckham's history of being a "gay icon", including frequent appearances in gay-focused magazines, and urged him to be the change needed in the football world.
He said: "This is a message to David Beckham. I consider you, along with Kim Woodburn and Monty Don, to be a gay icon. You're the first premiership footballer to do shoots with gay magazines like Attitude, to speak openly about your gay fans, and you married a Spice Girl, which is the gayest thing a human being can do.
"But now, it's 2022 and you signed a reported £10million deal with Qatar to be their ambassador during the FIFA World Cup. Qatar was voted as one of the worst places in the world to be gay. Homosexuality is illegal, punishable by imprisonment, and if you're Muslim, possibly even death.
"You've always talked about the power of football as a force for good, which suggests to me that you've never seen West Brom, but generally I agree. So with that in mind, I'm giving you a choice.
He continued: "If you end your relationship with Qatar, I'll donate this £10,000 of my own money - that's a grand for every million you're reportedly getting - to charities that support queer people in football.
"However, if you do not, at midday next Sunday, I will throw this money into a shredder, just before the opening ceremony of the World Cup, and stream it live on a website I've registered called benderslikebeckham.com.
"Not just the money, but also your status as a gay icon will be shredded. You'll be forcing me to commit a crime, although even then, I reckon I’ll get off more lightly than I would if I got caught [engaging in sexual activity with another man] in [Qatar capital] Doha. The choice is yours. I look forward to hearing from you."