The immune system has already been dealing with the problem of finite resources for the past 500M years or so.
The biggest issue is that circulating neutralizing antibodies take up a finite fraction of the blood. One of the reasons why neutralizing antibodies fade over time is to literally make more space for other antibodies. One of the reasons why you need a boost of a vaccine at least 6 months after the first dose is that somatic hypermutation takes time to develop high affinity neutralizing antibodies that are effective in much small fractions of the blood.
Even after that though neutralizing antibodies still tend to fade and what is left is just the memory T-cells and B-cells in lymph tissue. So the immune system naturally activates and deactivates. But this is more like a book on a bookshelf. It isn't really consuming "energy" and there's a massive amount of them that are available.
You also need to be much more worried about all the naturally occurring viruses and antigens that you're being exposed to--including the ones that don't actually get you sick at all. You're worrying about the sparring partners of a heavyweight prizefighter "sapping their energy" and ignoring the real opponents.
If you've read something about COVID and lympocytopenia ("T-cell exhaustion") then that only happens in severe COVID and shows up in autopsies and is studied as an indicator of how mild (unhospitalized) COVID turns into severe life-threatening COVID. The people who talk about that on social media are literally ignorant idiots who don't know what they're talking about.