Rizwan and Irfan both suffered hamstring injuries in the third match in Rawalpindi, where a second-string New Zealand side won by seven wickets to level the five-match series 1-1
His NOC was valid until the first week of August.
"We wish him all the best and it would be appreciated if his privacy is respected," said an Essex statement on Tuesday.
A source close to Tamim also said the same thing, referring to something 'very personal and urgent' to Tamim for his departure.
He will return to Bangladesh as soon as possible, the source added.
The 28-year-old opening batsman joined Essex for eight matches but is flying back having played just one match, against Kent on Sunday where her scored seven runs.