The claim is mostly false because the DNC did approve a resolution urging Congress to abolish ICE, but available reporting indicates the vote was not unanimous and that an earlier approval involved the Resolutions Committee rather than the full DNC.
The Democratic National Committee just unanimously approved a resolution calling for the abolition of ICE

DNC Votes to Back Eliminating ICE as Trump Expands Deportations - Newsweek
Source: newsweek.com
Key Evidence
ABC News reports that Yvonne Reeves-Chong voted against the abolition measure, disproving the claim of unanimity.
Newsweek, Yahoo News, and the Washington Times report that the DNC did approve an abolition resolution, while also describing disagreement over ICE policy and, in the Washington Times account, approval of a separate reform resolution.
Earlier Baltimore Sun, SignalSCV, Solwd, Epoch Times, and America First Report articles concerned the Resolutions Committee's action, not necessarily the final full-DNC vote.
What the Evidence Shows
The claim contains two separate assertions: that the Democratic National Committee approved an ICE-abolition resolution, and that it did so unanimously.
• Approval of an abolition resolution: Recent reports from Newsweek, ABC News, the Washington Times, and Yahoo News indicate that the full DNC approved a resolution calling on Congress to abolish Immigration and Customs Enforcement. The Washington Times additionally reports that the DNC approved two competing resolutions, one supporting abolition and another supporting reform.
• Unanimous approval: The available evidence does not support unanimity. ABC News specifically reports that DNC member Yvonne Reeves-Chong voted against the abolition measure. That directly contradicts the claim that the resolution passed unanimously.
• Decision-making body and timing: Earlier reports from August 13 and 14 described the DNC's Resolutions Committee advancing or approving the abolition resolution. Later reports published August 15 and 16 describe action by the full DNC, apparently approving both abolition and reform resolutions. Therefore, the committee-stage reports should not be treated as evidence of a unanimous full-DNC vote.
The most accurate formulation is that the full DNC approved a resolution urging Congress to abolish ICE, but the measure faced at least one recorded dissenting vote and was not unanimously approved.