Did John Dee design the Globe Theatre?

No.
John Dee (13 July 1527–1608 or 1609) was a noted mathematician, astronomer, astrologer, occultist, navigator, imperialist,[4] and consultant to Queen Elizabeth I of England. He devoted much of his life to the study of alchemy, divination, and Hermetic philosophy.

He did own some globes!
Dee promoted the sciences of navigation and cartography. He studied closely with Gerardus Mercator, and he owned an important collection of maps, globes and astronomical instruments. He developed new instruments as well as special navigational techniques for use in polar regions. Dee served as an advisor to the English voyages of discovery, and personally selected pilots and trained them in navigation.

https://en.wikipedia.org/wiki/John_dee

The argument of 'Theatre of the World' is that the design of the theater built by James Burbage, which influenced the later theater movement, including Shakespeare’s Globe, was based on classical theory on the design of the Roman theater as expounded in Vitruvius’s book on architecture. The arguments used in building up this theory emphasize the fact that Vitruvius was known among the artisan class in Elizabethan London through the teaching and writings of John Dee and could therefore have been known to Burbage.
http://www.nybooks.com/articles/archives/1970/mar/12/theatre-of-the-world/

James Burbage (or Burbadge) (1531–1597) was an English actor, theatre impresario, and theatre builder in the English Renaissance theatre. He built The Theatre, the facility famous as the first permanent dedicated theatre built in England since Roman times.
https://en.wikipedia.org/wiki/James_Burbage

The Theatre was an Elizabethan playhouse located in Shoreditch (in Curtain Road, part of the modern London Borough of Hackney), just outside the City of London. It was the second permanent theatre ever built in England, after the Red Lion, and the first successful one. Built by actor-manager James Burbage, near the family home in Holywell Street, The Theatre is considered the first theatre built in London for the sole purpose of theatrical productions.
https://en.wikipedia.org/wiki/The_Theatre

It seems to this novice that any connection between John Dee and the Globe Theatre is tenuous at best. Maybe he influenced James Burbage who built The Theater in 1576.

Interesting note to confuse things:
The Globe was built in 1599 using timber from an earlier theatre, The Theatre, which had been built by Richard Burbage's father, James Burbage, in Shoreditch in 1576.
https://en.wikipedia.org/wiki/Globe_theater