It's impossible to answer for every museum, because every museum is different, but typically an art or history museum that focuses on exhibitions will not have any replicas or reproductions. Replicas would take up storage space that could be used to house unique artworks and artifacts, and would be unlikely to be actually indistinguishable from the originals - so visitors would not get the same effect in viewing them. Museums also rely to some extent on the authenticity and rarity of their collections to bring in visitors: people will wait in ridiculous lines to see the Mona Lisa in person at the Louvre, despite the portrait being one of the most widely-reproduced portraits out there, because there is something that they get (or want to get) from standing in front of the original canvas painted by Leonardo da Vinci.

The one major exception is living history museums, especially those with a thriving community of artisans. "Living history" means that the interpreters dress in period costume and do something in a workshop, domestic, or farm situation. Living history museums require way more artifacts than regular museums, or at least more of the same artifacts (a farming museum might want to collect one bucket to represent all buckets of that type; a living history museum needs multiple buckets per building), and because the artifacts are generally being used, there's a danger of losing unique historic items to breakage or theft. So they will either purchase reproductions or have them made in-house. In the museum where I work - which has both exhibition and living history stuff going on - we've purchased replica pottery from Denise Keegan Carpentier and Connor Prairie (another, larger living history site) and textiles from Rabbit Goody (an expert with the best name); we also have someone working in one of our barns making buckets and tubs, a weaver, a broom-maker, and a bandbox maker (but unfortunately the focus of production here is for putting on sale in the gift shop rather than for fitting out the houses and barns).

This is a good and much more challenging question than it seems on the surface. As a rule, it's bad policy to allow forgeries and reproductions to stay on display unless they serve an educational function, in which case they are usually clearly labeled as reproductions.

In general practice at large national museums, what you see in the exhibits are authentic historic artifacts and works of art insofar as we know. Scientific and historic analysis of art and artifacts takes many forms, and like every other discipline new methods of verifying authenticity develop over time. Without detailed, recorded archaeological contexts or other forms of documentation (sale records, mention in historic texts, etc), it's often difficult to ascertain things like date, place of production, or, yes, authenticity. Even archaeological context can be faked or interpreted loosely to increase the value of the art or artifact, and artifacts can be wrongly associated with historical texts and other documentation. Our understanding of an object's origin can change over time as we develop new ways to study them, and sometimes that means we discover that an object in a museum isn't what it appears to be.

There are many documented cases of well-known objects in museum collections turning out to be fakes. This was especially the case when museums invested in conservation and research science departments and introduced materials analysis technologies, starting in the late 1970s and continuing into the present. To give you an example from my own research on medieval metalwork, there was one object in a museum in Baltimore that was used for years as a kind of "linchpin" in a number of historic arguments about this particular type of metalworking. In the late 1980s, samples were taken and analyzed and it was discovered that the chemical makeup of the object could not be medieval. It was a 19th-century forgery made for sale on the art market. This analysis forced historians in my field to drastically reconsider our understanding of this type of object. The object itself was removed from display and placed in the museum's storage.

Also in my field is a large body of counterfeit objects that were modeled upon authentic objects and share many characteristics with them. For years these counterfeits were on display in museums all over the world, including in the Louvre, the British Museum, and the Metropolitan Museum of Art. They are still on display in some smaller museums, despite being soundly debunked in the 1990s. I still receive questions about these fakes from museum staff from time to time, and they still turn up in high profile auctions described as authentic.

This is an ongoing challenge that museums face, and museums are constantly researching their own collections. Most are honest when they come across replicas and fakes, and some even display their fakes to educate the public on how our understanding of artifacts changes over time. For example, the Dumbarton Oaks Museum in Washington, DC, is home to a sculpture long believed to be Aztec. The iconic sculpture was used in exhibits, on book covers, and in many academic arguments. Over time, scholars determined that it too was likely a 19th-century fake. The sculpture remains on display in the museum with wall text discussing how our understanding of it changed over time.

I do have to disagree with the previous poster. Every museum I have ever worked with (and I have worked with very large collections worldwide) has housed TONS of fakes and replicas. They aren't usually on display, but they are very often kept in storage. They are used for education, as controls for the study of genuine objects, and sometimes even exhibited themselves. There are even "teaching museums" that display replicas prominently and use them for education in conservation, archaeology, and other types of hands-on study. The RGZM in Mainz is a great example of a museum that predominantly houses replicas and uses them in exhibitions.

In short, museums do their best to ensure the authenticity of the objects and artwork they display. However, we can't always be certain, and even replicas and fakes have their uses.

While I like and understand OP's question, "real deal" and not the real deal can have lots of different shades of meaning.

• So there is the 100% real item thats be tracked over time (provenance) and is undisputedly the item. Thats most common for individually famous items i.e., there is one Mona Lisa, one Picasso's Guernica, one hope diamond.
• Then there are things that are thought to be real but turn out to be fake. Currently there is a DiVinci called Salvator Mundi thats a hot topic of debate - it might be painted by Divinci, or one of his students and that affects the price and significance greatly. (Similarly, I actually just published on an item owned by a famous museum that is supposed to be from the 15thc, but I'm convinced is actually from the 1700s, still making it old, but not as old as the item is said to be)
• Then there are items that were absolutely real, but they got damaged a bit and someone fixed them, went a little nuts, fixed them too much and they hardly represent the original now. But the base item is "real". One example is Ecce Homo, which is that painting of jesus that someone "fixed" and it now looks like an awful monkey.
• Then there are various distinctions of copies. The ancient Romans remade Greek statues, so they're "real" ancient but they aren't the originals.
• Cast collections are exact duplicates of a sculpture (from casting with molds) that were popular in the 17th and 18th century and allowed people across Europe to study things found in other countries. These aren't real per say, but they aren't fake. They're also explained when they are on display as being casts.
• Similarly, artifacts under scientific study are often modern day casts (exact replicas) so you might see the same ancient human bones in multiple places (for example, Lucy bones are cast from the original, but the original stays under study in Ethiopia)
• Sometimes, for animal (dinosaur) skeletons, some of the bones are on display but over time, an incomplete skeleton has been found so there are real bones and hypothesized bones on a single model. Sometimes the not real bones are a different color or texture.
• For taxidermy, sometimes its just the hide that is "real" over plastic or metal forms. So the inside isn't flesh, but at the same time, the outside isn't synthetic.

There are a lot of grey areas when a layperson says "the real deal", which is understandable because most people aren't taught these complexities. Sometimes, the object labels will say what is "real" and what isn't but thats very dependent on how updated the museum is (which is in turn dependent on how much $ they have) and how much they want to go into details. Often words like "this cast of a dinosaur..." can give some clues.

In general, I would say that nearly everything in an accredited museum has a strong degree of "real" because the accreditation system wouldn't allow somewhere without integrity and too many fake things to be admitted. But there is a lot of gray area.