The short answer is no. The medium-long answer is only for commercial use. The longer answer is I am not a lawyer, but I'll take a stab at explaining this anyway — specifically for the USA.
The first question is, what is a CDL? A CDL is a Commercial Driver's License. The important word here is Commercial. CDLs are mandated for operation of a commercial motor vehicle by The Code of Federal Regulations (CFR) Title 49, Subtitle B, Chapter III, Subchapter B, Part 383, Subpart B, §383.23, "COMMERCIAL DRIVER'S LICENSE STANDARDS; REQUIREMENTS AND PENALTIES".
The second question is then obviously what is a commercial motor vehicle? That's defined in §383.5, "DEFINITIONS", which states that it "means a motor vehicle or combination of motor vehicles used in commerce to transport passengers or property", if and only if it falls into one of three groups, is designed to transport 16 or more passengers including the driver, or is used to transport hazardous materials. §383.91 defines those groups, the really interesting ones of which are "A" and "B". Buses towing less than 10,000 pounds fall into group B (if over 26,000 lb GVWR) or group C (if 26,000 lb GVWR or less) or into group A if towing more than 10,000 pounds.
So if it's not being used for commercial purposes, you do not need a CDL, full stop. But you may need some other special license or endorsement, and those requirements vary by state.