Does cleanse remove Zed's ultimate?

Cleanse does not remove the Death Mark ( Zed's ult) but the Quicksilver Sash does .Cleanse removes all disables and summoner spell debuffs but excluding Suppression

Cleanse doesn't remove the Death Mark from Zed but the Quicksilver sash does.

However it's an MR item and gives you a disadvantage if rushed against Zed.

Cleanse can still be useful though as it removes ignite.

The other answers correctly indicate that the summoner spell Cleanse does not remove Zed's Death Mark debuff.

However, they incorrectly indicate that Quicksilver Sash's active does.

In patch v6.9, Mercurial Scimitar (the item that Quicksilver Sash builds into and shares an active effect with) was changed to only remove crowd control effects, not debuffs:

The snapshot of the patch notes article by Riot explains the change:

The QSS and Mercurial Scimitar active has two uses: cleansing crowd control (Amumu ult) and purging combat debuffs (Zed ult). These cases differ in that crowd control is setup, while combat debuffs are the payoffs to such setup. When you cleanse CC, the enemy needs to lock you down again to aggress on you, but that tends to bpretty reasonable ask. By contrast, cleansing a combat debuff wipes the applicator's primary means of fighting you. So, those champs are pressured to snowball before their effects become meaningless, while their opponents are forced into buying QSS even if it's incredibly inefficient to do so. This leads to destructive effects on game balance which we've left unanswered for too long. With the understanding that there may be outliers we need to address, we're unshackling combat debuff champs from their quicksilver collars.

Zed's debuff is now no longer removable. Thus neither QSS nor Cleanse can remove it.