Ultimately the answer is "it depends". It depends on various things, including potential preprocessing and the distance you use (I guess Euclidean but be aware that this is not the only option). There are also various different methods of hierarchical clustering.
The major issue here in my view is the following, assuming you're using Euclidean distance. Ultimately the variables that you use define the meaning of the clustering (it is a wrong idea to think that there is only one "true" clustering of the data and one "true" selection of variables that should be found - there may be different clusterings in different groups of variables and different clusterings depending on how exactly you aggregate them). If you standardise the variables, they will in a well defined sense all have the same weight in your clustering.
Now the question is: If you have strongly correlated variables, in your situation, does this mean that essentially all these variables encode the same information, which should only be used once (i.e. with the same weight as any other single variable that is not highly correlated with others)? In this case it can be argued that proteins should be removed so that the information shared between them is only taken into account once, as should be. Note though that there are alternative methods such as replacing a group of highly correlated variables by their first principal component (which represents the "shared information" better than if one is used and the others discarded).
However it may also be that regarding your aim of clustering, if several variables are highly correlated, this in fact adds information that should be used by the clustering process. An example is that in social statistics wealth and education level may be highly correlated, but they are still essentially different aspects of what is of interest, so the clustering should use them both anyway assuming that both are relevant to the clustering aim. The correlation basically then means that they together have a strong influence on the clustering, which may well be appropriate as each may be important to the clustering aim in its own right.
It is important to realise that this is an issue that the data cannot decide on their own. It depends on the intended meaning and use of the clustering, and on knowledge about the meaning of the variables.