No you do not need to use cluster standard errors.
Cluster standard errors are for when observations are "clustered" within other observations at a higher level of analysis. So you might have a situation where students are nested within schools, or where you have multiple observations "clustered" within the same individual ("Sarah" has two observations in the dataset, one at time 1 and another a time 2).
It might SEEM at first glance like this also applies to your situation since you COULD think about people being "clustered" into different "groups" based on whether they got the treatment or not. But if you really were thinking about it that way then there would be no way you could include "treatment" as a variable in your model, since you would be considering "treatment group" as a way of identifying observations at a higher level (they way "person ID" or "name" identifies people) and not a variable that tells you things about people.
In your situation "experimental group" is not a way of identifying other types of observations but a variable - a characteristic of observations. People are assigned to either the treatment or control group in the same way they are either male or female. So there is no reason to include cluster robust standard errors in this analysis. It would be like including cluster standard errors for "gender" or "race" or "income level," which is obviously not something we usually do.