Companies: almost always yes
Article 51 of the Corporate Tax Law requires every taxable person to register and obtain a tax registration number. A company incorporated in the UAE, on the mainland or in a free zone, is a resident taxable person from its first tax period, whether or not it trades or makes a profit.
The common objection is 'we have no revenue yet'. That changes what you pay, not whether you register. A holding company in Abu Dhabi that owns shares and receives no income still has a registration obligation unless it is an exempt person under the law.