Above a link to an article about the fund hitting £1m, Dominic Peacock wrote "I've just donated", before making an offensive comment.
The East Riding of Yorkshire councillor later deleted it and said: "I shouldn't have put it on in the first place."
Council leader Stephen Parnaby said he would not tolerate such remarks.
Labour MP Mrs Cox died when she was shot and stabbed in Birstall, West Yorkshire, on Thursday.  A fund set up in her memory has received more than £1m in donations.
Mr Parnaby said: "Following inappropriate and offensive comments made on social media relating to the sad and tragic death of Jo Cox MP, I have suspended councillor Dominic Peacock from the East Riding of Yorkshire Council Conservative Group with immediate effect.
"I shall be reporting this to a meeting of the full group and my recommendation will be that he be expelled.
"This is not the time for inappropriate remarks, however intended, and I will not tolerate them in my group, and residents should expect their elected representatives to act in an appropriate and compassionate way."
He said he would be asking the Conservative group to make a donation to the memorial fund.
Mr Peacock posted the comment in the Vote Leave Beverley & Holderness, Haltemprice & Howden page on Facebook.
A spokesperson for the Vote Leave campaign also condemned the remark, saying: "These comments are appalling, abhorrent and we believe that he should be thrown out of the Conservative Party.
"He is not welcome to campaign with us."
Mr Peacock, who is the councillor for Minster and Woodmansey Ward, "apologised unreservedly" but declined to comment further when approached by the BBC.
His comments have also sparked a series of angry responses on social media.
Chevan Ilangaratne wrote on Facebook : "What an utterly odious and inappropriate comment to make following what was an absolute tragedy for mankind last week."
Jonathan Ware described the comment as "absolutely ghastly", while Tony Green called it "an absolute disgrace".
A Conservative councillor has been suspended over an offensive Facebook post he wrote mocking fundraising in memory of killed MP Jo Cox.