As a general rule, economically it's better to have rich trading partners than small ones. Imagine you are a mayor of a small struggling town, you're worried about the local economy, and a new person moves in. Would you prefer them to be an average Joe or a billionaire? The billionaire, right?: they can buy more, and then the inhabitants of the town can use that money to bring in goods and services. 

Well the East Germans weren't a bunch of billionaires. They were poorer, on average, than the West Germans and less productive.

The East German economy at reunification was quite small compared to the West German economy - East German GDP is estimated to have been about 10% of West Germany's, while by population East Germany was about 25% of West Germany. 

The Western German government chose a 1-to-1 exchange rate between the eastern Ostmarks and the western Deutschmarks, which came into force in July 1990. This meant a boost in spending power of East Germans, and thus a boost in demand for western goods which benefited West German producers in the short-term.  However, this was at the price of a large transfer from West Germany to East Germany, driving up West German taxes and government debt. The East German economy was drastically transformed (fairly consciously), and saw rising unemployment and an estimated 35% fall in GDP in the first two years after unification (but calculating GDP for a Communist state is highly doubtful - for example how do you adjust for quality?). In 1995, East Germany is estimated to have producing only 60% of what it consumed - with the difference being made up by transfers funded by West Germany. 

Public transfers to Eastern Germany were about 4-5% of annual West German GDP from 1991-1996 and caused a rise in taxes and German government debt. 

The West German economy grew quite weakly post 1992, in the late 1990s and early 2000s Germany was often called "the sick man of Europe", due to low economic growth and high unemployment rates. Which sounds bad for reunification. But, economies are complex things, there's certainly cases of major European countries' economies performing weakly for other reasons:

1.  France's economic performance from 1992 to 1997 was much the same as Germany's.

2. The UK was called the Sick Man of Europe in the late 1960s and 1970s.

Economic downturns can have many causes.  

The German economy came away again in the mid-2000s, though I shan't discuss that in any more detail due to the 20 years rule.

In summary, reunification was directly costly to West Germany and was followed by a number of years of economic underperformance, but it's not clear that the underperformance was caused by reunification.

Sources

Jennifer Hunt, The Economics of German Reunification: A slightly longer version of the entry prepared for the New Palgrave Dictionary of Economics, Feb 2006, http://www.rci.rutgers.edu/~jah357/Hunt/Transition_files/german_unification.pdf

Christian Dustmann, Bernd Fitzenberger, Uta Schönberg, and Alexandra Spitz-Oener, From Sick Man of Europe to Economic Superstar: Germany’s Resurgent Economy, Journal of Economic Perspectives—Volume 28, Number 1—Winter 2014—Pages 167–188, https://pubs.aeaweb.org/doi/pdfplus/10.1257/jep.28.1.167

 Fabio Canova, Morten O. Ravn, The Macroeconomic Effects of German Unification: Real Adjustments and the Welfare State Review of Economic Dynamics 3, 2000,

http://apps.eui.eu/Personal/Canova/Articles/maceffer.pdf

Demetrios Giannaros, *Twenty Years After The Economic Restructuring Of Eastern Europe: 

An Economic Review*, International Business & Economics Research Journal – November 2008, ungated copy https://www.researchgate.net/publication/237597889_Twenty_Years_After_The_Economic_Restructuring_Of_Eastern_Europe_An_Economic_Review/download