Can I enter Dubai airport visa-free if I have a valid H-1B visa?

Courtesy of Emirates web site sharing of the Timatic tool, the information used by airlines for passenger entry requirements (and as noted in the deleted @HankyPanky response), you would not require a visa (bold italic added emphasis mine):

Destination - United Arab Emirates (AE)

Passport
Passport required.

Document Validity:
Passports and other documents accepted for entry must be valid for a minimum of 6 months from the arrival date.
Passports and other documents accepted for entry issued to passengers who stay less than 3 months must be valid for a minimum of 3 months from the arrival date. This does not apply when entering in Abu Dhabi (AUH).
Admission and Transit Restrictions:
Admission refused to passengers with a hand-written passport at Dubai (DXB). Additional Information:
Passengers with dual nationality must present the same passport at immigration upon entry and departure.
Warning:
Visitors with an extended validity in their passports are not allowed to enter.
Visa
Visa required.

Visa Issuance:
E-visas can be obtained before departure at www.ednrd.ae . Passengers must have a printed e-visa confirmation. Airlines can check the validity of the e-visa on the same site by clicking on "Query GDRFA-D APP".

Nationals of India with a normal passport valid for a minimum of 6 months from the arrival date with a visa issued by USA, United Kingdom or a Schengen Member State which is valid for a minimum of 6 months can obtain a visa on arrival for a maximum stay of 14 days. They can apply to extend their stay for an additional 14 days.

Passengers with a normal passport and a confirmation that a visa has been approved before departure can obtain a visa on arrival for a maximum stay of 96 hours, 30 days or 90 days. Passengers must have a passport valid for a minimum of 6 months from the arrival date. The visa approved before departure must be deposited by the sponsor at the Immigration Authorities.

• Deportation on the same flight will follow if sponsor (or representative) is not present at the time of passenger's arrival at the airport.

Additional Information:
Valid visas in expired passports are accepted provided accompanied by new passport.

Warning:
Visitors not holding return/onward tickets could be refused entry.