The only way the Treaty of Versailles actually contributed to the downfall of the Weimar Republic is because the Germans used it as a scapegoat, not because it actually was particularly punitive.

The "Deutsches Reich" (which was the real name of the Weimar Republic) was particularly incensed by three key factors: firstly, the so-called "stab in the back," where it seemed that a nearly triumphant Germany utterly collapsed in a matter of months, ostensibly by speculators (particularly Jews), the idea of German responsibility for the war, which (despite being true) was heavily resented by the average German: and lastly, the rise of Poland out of former German territories. Never mind that the population of those territories was primarily Polish, that the Germans had gone out of their way to antagonize the Poles (in the case of Silesia and Posen), and that the Germans had resurrected the Polish state in the first place (as part of Brest-Litvosk).

These factors, combined with the arms limitations, meant that a large section of the population, primarily former soldiers and nobility, became extremely reactionary. This was not helped by attempted communist uprisings that were put down brutally by both German troops and paramilitary forces known as Freikorps. Incidentally, the forced disarmament of Germany was a massive boon to the government budget, so any excuse that Germany could not afford the reparations payments (which had already been diluted to a third of the face amount by the Entente powers prior to the signing) is ludicrous.

These factors led to the German government attempting to sabotage their own economy to renegotiate the reparations. During the period known as Weimar hyperinflation, rather than do something sane like raise taxes or cut spending, the German government chose to print marks and use the marks to buy gold for reparations, thus rapidly inflating the currency and destroying the economy. This coincided with a refusal to properly pay the reparations, prompting French and Belgian troops to move into the industrial Ruhr region to force Germany to satisfy payment. But in response, Germany declared a general strike that further crippled the German economy but in return made the Franco-Belgian occupation unprofitable (and also started another communist uprising). In effect, Germany decided to be the petulant little child that whines that he's hungry and throws his food on the floor because he wants ice cream.

This did cause the Entente to further renegotiate Germany's reparations, in what became known at the Dawes Plan. And the reactionaries were still upset about it, considering the deal to be treason. Germany still did not consistently make its reparation payments, albeit with slightly more regularity.

As the German economy revived, a loose political alliance between labor activists and social democrats along with moderate industrialists had been formed, called the Grand Coalition. This political alliance, while it made great strides for reviving the German economy, ended up neglecting the interests of the middle class. This demographic ended up supporting the National Socialist party, although they never reached a majority. Even during the Great Depression, which is often cited as a reason for the rise of the Nazis, the Nazis never got more than 35% of the vote.

However, what did happen is that the Grand Coalition fell apart. In its place, the reactionary party, the German National People's Party, made an alliance with the Nazis in order to have a majority in the Reichstag. The leader of the DNVP, Von Papen, thought he could manipulate the Nazis into supporting him. He was wrong. Thanks to a certain article in the Weimar Constitution granting emergency powers to the Reich Chancellor, Hitler was able to seize complete power after starting a fire at the Reichstag that he blamed on communists. And that was the end of the Weimar Republic.

In effect, Germany was upset that she had to take any responsibility for her actions during World War I, proceeded to blame everybody else for her problems, and attempted to immolate itself in some twisted economic version of Wagner's Gotterdammerung in order to satisfy a thoroughly radicalized extreme right-wing faction that refused to cooperate with anyone unless they got what they wanted.

Sources:

Gatzke, Hans: Stresemann and the Rearmament of Germany

Fergusson: When Money Dies: The Nightmare of Deficit Spending, Devaluation, and Hyperinflation in Weimar Germany. (I suggest ignoring the political views here, but the historical facts are fairly readable without getting too deep into fun economics)