How to prevent an IP from displaying in a live capture?

You can apply a display filter like !(ip.addr == ip.add.re.ss1) and !(ip.addr == ip.add.re.ss2) during live capture.

You can build the display filter expression step-by-step by right-clicking on a line representing a packet field (like source IP address) in the packet dissection pane and choose Apply as Filter -> ... and not Selected from the context menu. To do so conveniently when a capture is running, it is better to switch off the automatic scrolling of the packet list before doing it.

You cannot build display filter expressions which use pseudo-fields (such as ip.addr which represents ip.src and ip.dst simultaneously) this way (i.e. Apply as Filter ->) because they are not available as lines in the packet dissection, but you may use e.g. ip.dst to get the address to the filter expression and then manually change ip.dst to ip.addr.

You can save named (labelled) pre-defined display filters for single-click application in future: at the rightmost end of the line which contains the display filter form field, there is a "+" button. When you press it, another form line is inserted between the original one and the packet list pane, where the filter expression is pre-filled with a copy of the currently used one, and it is enough to fill in the "label" form field and press OK. The additional line disappears and a button with the label you've just filled in is added to the right from the "+" button. Pressing one of these "label" buttons applies the corresponding filter. There is also another button, Filter Expression Preferences, on that additional line, which is a shortcut to the preferences dialog, where you can enable/disable, add and delete your single-click display filters. Only the enabled ones are available as buttons next to the "+".