One important thing to understand is that many of the best historic site to visit around Dubai are going to be outside the actual city. Dubai is an incredible story of how rapidly change can occur in the modern ages, as it was an incredibly small and sleepy desert village as recently as the 1950s. In other words, depending on what you're looking at in Dubai, a city that is the definition of designed for the modern crowed, you'll have to be a bit fluid on your definition of "historic."
That being said, there are several places in and around Dubai that should pique the curiosity of visitors who want to see what the older history of the area looks like.
Old Town is the part of the city that most reflects the everyday life of many of the long-time residents of the area, with traditional souks, markets, and even some museums celebrating the recent past and meteoric rise of Dubai. This is a place where you can experience local cuisine, local markets, and enjoy the wonderful experiences of a local Arabian Tea House or local art. This is a great place to take a look at the past and culture of the area.
Jumeirah Archaeological Site
Located outside of Dubai, the Jumeirah Archaeological Site is one of the most popular sites in the entire United Arab Emirates (UAE). Originally built sometime between the 9th and 11th centuries. There are only some scattered ruins left but you can visit the area to get a sense of awe and wonder of the market and trade routes that went through there over 1,000 years ago while you can follow this up with a trip to the Dubai Museum to see the excavated collections that originally came from this site.
Village of Hatta
Hatta is a mountain town that is nestled in the Hajar Mountains. This is still an active village, one with plenty of ruins from a more militant medieval time that included forts and citadels. The Heritage Village within Hatta is a place where you can see rooms, an old fashioned courtyard, local jewelry, clothing, metal work, pottery, and more. Guard tower still stand in the Hatta Fort, and it's an impressive place to visit to get a little taste of living history.
The Hatta Dam is a modern construction that created a reservoir and creates a sustainable water supply for the village's population. The combination of old, very old, and modern makes Hatta a remarkable place that is definitely worth a visit.
The Saruq Al-Hadid Archaeology Museum is just the place for a history fan to scratch that itch when feeling a bit overwhelmed by the modern. Focusing on the iron age history of the area, and featuring a wide array of artifacts from the famous and recently discovered site at Saruq Al-Hadid, this site was discovered from a traveling Sheik who noticed a massive color change in the sand while flying into Dubai, and further investigation led to one of the largest sites in the UAE. This is a hands-on museum that allows for plenty of interaction and a memorable learning experience.