There are many places for Afternoon tea in Dubai, not just the famous Burj Al Arab. Many other hotels also offer traditional English High Tea all over the city. The price tags are a lot lower than at Burj Al Arab, but on other hand at the Burj you sort of pay for the nice view of Dubai coastline.

The Ritz-Carlton
Al Mamsha Street, on the Walk of Jumeirah Residence and close to Dubai Marina.
Afternoon Tea at The Lobby Lounge
English high tea with sandwiches, baked scones, Devonshire clotted cream and assorted pastries served with premium tea of your choice.
Hours: 14:00 – 16:00
Price: AED 150 per person
Jumeirah Zabeel Saray Hotel
Crescent Road West on The Palm Jumeirah.
Afternoon tea at Sultan’s Lounge
Here you’ll find both a traditional English Afternoon Tea menu and one Turkish inspired high tea with Ottoman theme.
Hours: 15:00 to 18:00
Price: AED 155 per person
Desert Palm Dubai
Al Awir Road, southeast of the International Airport.
Enjoy traditional English Afternoon Tea made of homemade scones, with cakes and fruits presented in classic three-tier style while watching a polo game. Polo matches are played between October and April and this time you can be served on their outdoor terrace.
Hours: 16:00 to 18:00
Price: AED 140 per person

Park Hyatt
Baniyas Road, along Dubai Creek.
There is a comfortable ambience overlooking the Creek and Marina at Park Hyatt. They serve traditional afternoon tea.
Hours: 14:00 – 16:00
Price: AED 190 per couple

TGW Tea Salon
Located in the Dubai Mall, on Ground Floor opposite of Waterfall Atrium in Downtown Dubai.
The TGW Tea Salon serves a grand variety of tea with scones and patisseries in varoius forms, such as classic Afternoon Tea. Perfect stop during your shopping.
Hours: Sunday – Wednesday: 10:00 – 22:00, Thursday and Saturday: 10:00 – 00:00
Price: AED 42 – 80