Summary: You believe the Russians are evil and so feel there is no need to refute their propaganda at all. Others with different political beliefs may however choose to believe it unless they are refuted with facts and reassurances from their political leaders.
All the answers here are based on the underlying assumption that "Putin / Russia always lie". Some of the readers may have missed that this kind of framing itself is an obvious form of propaganda. In fact, if you question that basis itself, you get a more nuanced understanding of the politics at work.
An important thing to understand is that political propaganda survives on two important elements - inherent political bias and some underlying basis of truth.
Political bias in a person develops out of their own political experience and / or is created by a trusted political source you look up to. A good and very common example of political bias is nationalism, where in a citizen often believes their own country's political action as unquestionably right. (A famous political saying on this is, "My country, right or wrong!").
A good propaganda is targeted at an audience with a specific goal, based on an underlying truth in it that feeds into the political bias of the audience.
To understand this, let's consider the three propagandas cited in the question, and consider how it helps the Russian, and why western politicians had to refute it publicly and vociferously.
Ukraine was making nuclear weapons in Chernobyl
(Note that the actual propaganda was that Ukraine is attempting to make a "dirty nuclear bomb" - a dirty bomb is a bomb made with highly radioactive nuclear waste that causes radioactive pollution when detonated - and that they plan to explode this in Ukraine or Europe, and blame Russia.)
Goal: To create anxiety among western politicians and the citizens, and cast doubt on Ukranians.
The political bias (that this seeks to reinforce is): (1) "NATO backed Ukranians are evil / untrustworthy". (2) Russians are victims of a western + Ukranian conspiracy.
The underlying truth (on which this propaganda is based): Ukraine has a lot of nuclear reactors (and thus access to radioactive elements) and Ukranian military has experience in managing Russian nuclear weapons, and thus has the know how to make a "dirty nuclear bomb".
Ukraine had biolabs working on new biological weapons
(Similar reasons as 1).
Ukraine's government is "fascist" and "full of nazis"
Goal: Create support for the Russian military and its action. Cast doubt on NATO and Ukraine.
The political bias: NATO will always support and use anyone who is against Russia.
The underlying truth: Ukraine as a society is politically divided which is evident in the fight between the supporters of the western-backed right-wing Ukranian politicians and the Russian-backed left-wing Ukranian politicians. The Right-wing elements of the western-backed Ukranian politicians did start targetting Russian cultural identity in Ukraine and attacking the other side in an attempt to create a monoculture in Ukraine.
Now, if you believe that everyone in the west has the political belief that "Russians are evil and their words cannot be trusted", a western politician can confidently dismiss these propaganda as "typical Russian lies". But that isn't the reality - in both the United States and in Europe there are a many who do not accept that Russia is evil or that Ukraine is without blame. Or those who remain sceptical because of the underlying truth based on which the propaganda is based.
(While I have mentioned only US and Europe, note that this is more true among the non-western countries, and that is why Ukraine hasn't received much international support from them).
In such a political scenario, a western politician has no choice but to address and attack these claims publicly, to ensure that the propaganda doesn't find support among its own citizens, and in the international community. Moreover, note that the underlying fact in a propaganda gives some credence to a propaganda and so it needs to be countered with complete / more facts.
(I have chosen to treat all these claims in the question as a propaganda, just as the questioner did, to make it easier to answer the question. But note that another form of propaganda is to often call something politically true as a propaganda too. For example, I personally do not believe that the 3rd political claim is without merits - while the claims of the Russians that the current Ukranian government is full of Nazis is obviously exaggerated, actions of the Ukranian politicians to use identity politics to divide the Ukranians show a strong under-current of right-wing politics with a propensity for violence against the "others".)