Research administrator here from an R1 university (in the US). I've worked with labs in pre- and post-award for 13 years. Here's what I can tell you about working with researchers who learn admin tasks (like proposal writing, finance work, contracting, etc.). If the postdoc's intention is to be a PI -- there is no "PI School". This is an academic's only chance to learn these things before that work is "assigned". Once it's assigned, administrators assume you will know how to do things. This is not the case for postdocs and students. In my experience working with hundreds of my peers, we have some pity on the fact they are asked to do things they are not responsible for and will be elated that they take an interest in our work. We know that most groups cannot afford lab managers, and some PIs do expect postdocs to help shoulder this burden. In my experience, this is very normal in large labs, and in particular the natural sciences.
My advice for any postdoc or student is to learn how the basic components of the research actually work. You don't have to become friends with the administrators in your department, but it can have tremendous benefits. When I worked in a Chemistry department, I would have students come and bring me tea and ask me to help get them out of jam by ordering supplies on rush. They came to me when the dry ice didn't arrive on time. When the copier jammed, they knew I was the one to help. How to tell if the fax went through? Ask me. When they wanted to put in fellowship proposals, they came to me and I taught them how to pace their work. I taught them as much as I could about how my world works, and when I left that job, I received an outpouring of departing words and gifts primarily from students. I have never forgotten them, even though I left that job 8 years ago.
The main complaint I have about researchers I've worked with is that they tend to assume "admin work" is a waste of time. It's an attitude that is pretty insulting to me and my peers, who dedicate our lives to this work. Ideally, this should be a partnership. I'm here to support the research, help me help you. If you want your supplies to arrive on time, please don't come to me at 4:45pm on a Friday and expect me to put it through the system. Be thoughtful and plan ahead. Know who the approvers are and be kind to them. Our most effective students and postdocs do better in their research because they know the staff and get more out of the experience. It's an investment.
I'm working with a postdoc on a K99 application right now, and I take the time to make sure they are more likely to get that award--I am going out of my way for this person because they take the proposal process seriously and do their part, respecting deadlines. The central office deadline is Friday, but I'm going to ask to get her an extension until Tuesday. I do that because of her performance with administrative tasks. I am trying to get another postdoc to file their IRB paperwork, and it has been an extremely difficult process. The attitude of that postdoc is that all of this work is my work, and so I am not going to put myself out to help him. He is jeopardizing his own research by not digging into the paperwork as I asked, continuing to delay approvals on his research.
When I get assigned a junior PI, I can tell immediately if they have "it". When they pay attention to admin policies, work with me, their proposals get funded. The ones who always put "admin" off until the end, like an afterthought, usually lack project management skills. In the end, university research requires strong project management skills. You need to learn who your stakeholders are and manage them. I entirely agree that there will be nothing on your resume that shows one's awesome administrative skills. However, in my experience, those who are most effective in their research are the same ones who come to me with problems and the understanding that I spend my time on a solution. It's the partnership with administration that makes them effective.
All this being said -- if the postdoc's work is actually slipping -- they can't keep up with the basic research, there is a problem. Make sure the administrative work is being done effectively -- find good staff members in the department and ask for help. Learn the policies and procedures to avoid wasting time. If it's still too much, then yes, it's time to lower the amount of burden.