I think so, yes, but not always.
I go to a newly-qualified Russell Group university that joined in 2012 alongside Durham, York, and Exeter. So, prior to then, these universities hadn't been RG classified, which really proves how 50/50 the Russell Group actually is. Durham, for example, is an amazing university, but are you telling me that people would have shunned it prior to 2012? No. People have always thought highly of Durham, whether or not it was RG.
Universities such as Leicester and Kent are arguably on the up, both of which are not RG. They are still good universities, despite not belonging to the RG.
I do think that universities can be influential in terms of how employers perceive you (they know that RG universities have a reputation for academic excellence, for example), but they also know that achievement is also a personal quality. People go to Oxbridge and get third class degrees, and then some people will go to Bolton and get a first. Arguably, a first from Oxbridge is a lot harder to achieve than a first from Bolton, but it is a first nonetheless.
Employers may take your university into consideration, but they are more interested in you as a person.