Now this sort of thing can only be done with so-called 'autographic' works, ones of which there can only be one original. According to the theory of Nelson Goodman in Languages of Art this cannot be done with so-called 'allographic' works such as music, dance and theater where the history of the production of the work is not essential to the value of the artwork. There can be hundreds of copies, both written and printed, of Beethoven's Fifth Symphony and thousands of performances and they can all be authentic as long as they follow the specifications of the score. In this theory it is inherently impossible to forge a Beethoven symphony even though you might be able to forge a manuscript copy of one.
However, it is certainly possible to create a parody of a work by Beethoven. Imagine a musicologist, a composer and a manuscript forger getting together and writing a new composition so closely imitating the style of Beethoven that it could fool not only listeners but also professional musicians and other musicologists. Once written, the score would be handed over to a forger who would create, on old paper and with old inks, an exact facsimile of a typical Beethoven manuscript, scribbles and all. This could then be announced to the world and given a big premiere. This is highly unlikely for many reasons. First of all, there wouldn't be the millions of dollars of potential profit enough to attract people good enough to bring it off. Second, we have a pretty extensive knowledge of Beethoven's life and it would be difficult to find a niche big enough for a whole symphony to have been composed with no clues in the biographical material.
A work like this is called a 'parody' and many composers have done parodies of other composers' works throughout music history, often including some or all of the original work. For example, the concerto for guitar and orchestra by Joaquin Rodrigo titled Fantasia para un gentilhombre, written for Andres Segovia, uses themes from the 17th century Spanish composer and guitarist, Gaspar Sanz. Here is the original of one of the movements, the Canarios:
And here is Rodrigo's version for guitar and orchestra:
Of course, Rodrigo has added a great deal, but it is all based on the original themes. This practice goes back a very long way. In the 16th century the vihuelista Luis de Narváez did a parody of an original piece by Josquin des Prez called "Mille Regretz". Here is the original:
And here is the version by Narvaez which he called "Cancion del Emperador":
What Narvaez has done is take the whole structure of the music and transcribe it for vihuela (an instrument similar to the guitar and lute). To compensate for the short sustain of the plucked strings, he has added suitable ornaments dividing long notes into shorter ones.
Occasionally composers do something a bit nefarious when they take music from another composer and pass it off as their own. In 1783 Mozart took a symphony by Michael Haydn, revised the wind parts throughout and added a slow introduction to the first movement. He then performed it in a concert along with his Symphony No. 36, where it was undoubtedly accepted as his own work. That most of the symphony was actually by Michael Haydn wasn't discovered until 1907. Here is that first movement:
From about 1:24 on, it is Michael Haydn, not Mozart. Here is the original by Michael Haydn:
There are lots of instances of music being attributed to a famous composer that is actually by someone less well-known. This is an instance of a famous composer attributing music to himself by a lesser composer!