Yes, most certainly.
The UN (though, not a country, a good starting example) officially doesn't recognize Jerusalem as Israel's capital:
The United Nations General Assembly does not recognize Israel's proclamation of Jerusalem as the capital of Israel, which is, for example, reflected in the wording of General Assembly Resolution 63/30 of 2009 which states that "any actions taken by Israel, the occupying Power, to impose its laws, jurisdiction and administration on the Holy City of Jerusalem are illegal and therefore null and void and have no validity whatsoever, and calls upon Israel to cease all such illegal and unilateral measures."
And China, for example, does not (same source):
China recognizes East Jerusalem as the capital of Palestine.
On the other hand, some countries do recognize Jerusalem as the capital of Israel. The United States under President Trump controversially did this as recently as the end of 2017:
The warnings come as President Trump considers moving the U.S. embassy from Tel Aviv to Jerusalem, effectively recognizing the ancient city as Israel's capital despite Palestinian claims to East Jerusalem. The move could come as early as Wednesday, according to media reports.
So, it varies widely by country.
As for a comprehensive list of which countries have which stand, I couldn't find one. But know that a large number of countries don't, as shown by these 128 countries that voted against the United States' decision to do so.
As for the Palestine Authority, they recognize Jerusalem as their capital, not Israel's, hence why there is a lot of conflict:
Earlier this year, the Palestine Liberation Organisation's (PLO) Central Council, the second-highest Palestinian decision-making body, has recommended revoking recognition of Israel until the latter recognises the State of Palestine in its 1967 borders, with East Jerusalem as its capital.
Something I forgot to address earlier: if countries don't recognize Jerusalem as the capital, I believe your title asks what city would they be recognizing?
This is very unclear from everything I've seen, but it appears to be Tel Aviv (emphasis mine):
On December 6, 2017, US President Donald Trump announced the United States recognition of Jerusalem as the capital of Israel2 and ordered the planning of the relocation of the U.S. Embassy in Israel from Tel Aviv to Jerusalem.2
This suggests to me that the other city would be Tel Aviv.