The vote requirements for Continuing Resolutions is no different than other bills that may be introduced to fund the government (Budget Reconciliation is the exception).
In the House, a simple majority.
In the Senate, a vote of 60 (three-fifths) is needed to end debate (cloture vote). Following that, a simple majority.
For the 117th Congress (2021), the Continuing Resolution is H.R.5305.
H.R.5305 - Extending Government Funding and Delivering Emergency Assistance Act
It is known as a continuing resolution (CR) and prevents a government shutdown that would otherwise occur if the FY2022 appropriations bills have not been enacted when FY2022 begins on October 1, 2021. The CR funds most programs and activities at the FY2021 levels with several exceptions that provide funding flexibility and additional appropriations for various programs.
On September 21, 2021, the House passed the bill by a vote of 220 to 211 -- a simple majority.
The bill was sent to the Senate.
On September 27, 2021, the Senate voted On Cloture on the Motion to Proceed (Motion to Invoke Cloture: Motion to Proceed to H.R. 5305 ). This required a vote of three-fifths in favor. The vote failed 48 to 50.
Senate Majority Leader Schumer (D-NY) voted Nay so that the motion can be reconsidered later.
On September 28, 2021, Senator Schumer offered, separate from H.R.5305, S.2868 - A bill to temporarily extend the public debt limit until December 16, 2022. Unanimous consent was requested for passage of the bill. Senator McConnell objected.
On September 29, 2021, Senator Schumer stated that "Senate Democrats will be introducing a continuing resolution that keeps the government open until early December, ..." .
On September 30, 2021, Senator Leahy offered S.Amdt.3830 to H.R.5305 "in the nature of a substitute". Republicans offered three amendments to S.Amdt.3830. Debate and voting on the amendments was done "under the previous order" which limited debate time but applied a "60-vote" rule for passage of the amendments. The Republican amendments failed. S.Amdt.3830 passed 65-35.
The amended H.R.5305 was returned to the House where it passed 254-175. President Biden signed the bill.