Fate of antigen-presenting cells after antigen presentation to Helper T cells
In Janeway it's being said without further explanation (I will edit and add reference)   that B-cells and macrophages &quot;become targets&quot; of cytotoxic T-cells after these killer cells have become activated, have been primed. However, this answers your question only if presentation on MHCII (not I) implies that presentation on MHCI has occurred as well. Only then APCs that present to T helper will be targeted by cytotoxic T cells and die. According to my basic knowledge that is the case with Dendritic cells and Macrophages. According to the explanatory text of your question Macrophages only receive signals to kill pathogens they include. Your inference may be correct that they cannot become target of T-cytotoxic cells. It is beyond your question to ask if Macrophages always do get that signal. If no that puts them into the same category as Dendritic APCs.
Understanding that there exists a signal to survive (destruction of phagocytoses pathogens) leads the way to understanding that there is cell death without killing by Tc cells:  cells that present antigens to T helper cells will &quot;apoptotically&quot; die because they have been infected, if not killed by immune cells to prevent the production and shedding of virions. According to my basic knowledge it is coherent  that a cell presenting is a cell sentenced to apoptosis. Killing by T cells is killing at an earlier stage only.
However, I found no reference that explicitly confirms that - maybe with very few exceptions -  that B cells do present on MHCI, to the contrary I found reference - I promise to add this one link quite soon - that explicitly states that B cells only endocytose debris, &quot;protein&quot; (in the sense of protein vaccination) i.e. isolated antigen, not whole virous. Thus, any antigen they present would be on MHC II, not I, as the antigen is not &quot;endogeneous&quot;, not produced by the B cell as it has not become infected. This inference from basic textbook knowledge is supported, in this context, by the fact that B cells, just as T cells - which they are thought of as of one and the same category, the Lymphcytes - should not be killed after presentation (whereas T8, cytotoxic T cells do not present at all) as parallel to killing of T cells they still have their job to do: presentation activates the B cells which start to divide and produce antibody.
Consequently, it matters if a B cell not presenting on MHCI, thus not being killed by Tc cells, is sentenced to apoptosis and cell death because it has engulfed the antigen. In my opinion B cells - and I will add any reference to the contrary - not even need, as Macrophages, a signal to survive, as they cope with antigen alone and there is no interference with their regular by viral infection. (However cells hosting active HIV produce protein for viral ends for a long time before they finally die, but this is different.)
To sum up:

Dendritic cells present on MHCII and I, hence killed  by Tc cells or dying by apoptosis

Macrophages present on MHII and I, but according to textbook different to Dendritic cells have the option or are programmed to receive a survival signal to consume the pathogens they harbour

B cells present on MHCII and, according to my understanding: maybe, on MHCI, which would put them too in category 1: same fate as Dendritic cells, however basic knowledge coherently applied speaks in favor for B cells regularly not presenting on MHCI, not getting killed by Tc cells and not undergoing apoptosis as not &quot;infected&quot; by whole virus


So my own personal answer on B cells based on basic textbook knowledge and lack of explicit statements to the contrary might be very different from the answer regarding dendritic cells (and macrophages, see explanatory text of your question).
The B cells do not die after presenting to T Helper cells.