I have the following code which works as expected except, as noted in the code, when a macro begins a line of the xltabular being built:

documentclassarticle

usepackage[papersize=5.5in,8.5in,margin=0.6in,bottom=0.7in]geometry
usepackagearray
usepackagemulticol
usepackagexparse
usepackagexltabular
usepackageetoolbox

%% https://tex.stackexchange.com/questions/487572/obeylines-and-gappto-from-etoolbox
%% Collect the body of the xltabular in tabbody:
begingroup
 lccode`~=`^^M
 lowercase%
endgroup
 deftabline#1~%
 xapptotabbodyunexpanded#1\hline~
 


makeatletter
NewDocumentEnvironmentlistit%
 gdeftabbody
 noindent
 beginminipagetextwidth
 raggedcolumns
 beginmulticols2
 col@number@ne
 mathchardefLT@end@pen=0 %
 begingroup
 offinterlineskip
 everypar=tabline
 obeylines
%
 endmulticols
 endminipage

makeatother

NewDocumentCommandexplain%
 endgroup
 vspace-baselineskip
 beginxltabularlinewidthX
 hlinehline
 tabbody
 hline
 endxltabular


begindocument

beginlistit
Lorem ipsum dolor
sit amet, consectetur
adipiscing elit
sed do eiusmod
tempor incididunt
bfseries See Note
ut labore
et dolore
magna aliqua.
Ut enim ad
explain

Note: verb+bfseries+ by itself is ignored. verb+textbfSee Note+ produces verb+Extra fi+ error.
endlistit

enddocument

Using leavevmodebfseries See Note produces the expected output. Why is this necessary?

What am I missing?

Diagnosis

The problem is that the lines of your tabbody are collected by the tabline macro, which is inserted by everypar. However, the everypar tokens are only inserted when a paragraph starts, and bfseries used in vertical mode doesn't cause TeX to start a paragraph (i.e., to switch to horizontal mode). So, when one of the special lines designed to be read by tabline starts with bfseries, TeX is still in vertical mode when it finds the bfseries (this is right after an end-of-line has been turned into par). It expands it, and only later when a 〈horizontal command〉¹ causes a switch to horizontal mode, it inserts tabline which will collect the rest of the line. But when this happens, it is too late, the bfseries has already been fully expanded and digested, it is not grabbed as part of the #1 by the inserted tabline.

Example: suppose that TeX is collecting these lines:

tempor incididunt
bfseries See Note

tempor incididunt is the first argument of the tabline macro that was inserted by means of everypar when the paragraph was started with the t from tempor incididunt (t is a 〈letter〉 and therefore a 〈horizontal command〉; when TeX finds it in vertical mode, it switches to horizontal mode to start a new paragraph). So, the expansion of this tabline consumed tempor incididunt• from the input stream, where • represents an active end-of-line character (it is active because obeylines was used above). After these tokens have been removed from the input stream, the replacement text of tabline is inserted, with tempor incididunt substituted for #1:

xapptotabbodyunexpandedtempor incididunt\hline•

Once xappto has been fully processed, TeX finds the • that was just inserted, which has been made let-equivalent to par by obeylines:

obeylines:
macro:->catcode `^^Mactive let ^^Mpar 

Therefore, when this • token is digested, TeX ends the paragraph and switches to vertical mode. The next token from the input stream is bfseries. This is a macro, therefore it is expanded. I pass on the details of its expansion (protectbfseries where the second bfseries has a space at the end of its name, etc.). What is important is that this bfseries doesn't start a new paragraph (its expansion doesn't contain any 〈horizontal command〉). This only happens when TeX digests the S from See Note, i.e., after bfseries has been fully processed. At this point, TeX switches to horizontal mode because the S is a 〈horizontal command〉; it inserts the indentation box (invisible here because parindent is 0pt), then the tokens stored in everypar, in your case a single tabline, and resumes normal processing of the input stream. The just-inserted tabline token is expanded, this grabs See Note as the first argument (you see, bfseries isn't present in this argument, it is already behind us), consumes the active end-of-line character that follows (because it is part of the 〈parameter text〉 of the macro), then the replacement text is inserted with See Note substituted for #1:

xapptotabbodyunexpandedSee Note\hline•

(note that this xappto call doesn't append the bfseries to tabbody, it has definitely been lost for tabbody) and the process goes on, as we just explained.

Proposed solution

I propose not to rely on everypar. Rather, we can make the end-of-line character active in the innermost group of your listit environment definition and redefine the active end-of-line character to be let-equal to tabline (see my listit@obeylines macro). This way, every end-of-line character in the special portion becomes equivalent to a tabline token and is expanded right before the next line. This way, its #1 grabs everything until the following end-of-line, and thus no command is lost.

This method even allows you to have par tokens among the text that is collected in tabbody (see Paragraph break here:par But... in the example below). Of course, it requires a means to stop the special collection process. Given your example, I decided that a line starting with explain marks the end of this process (cf. listit@checknext). Of course, one could use a different end marker if you wish.

documentclassarticle
usepackagemulticol
usepackagexparse
usepackagexltabular
usepackageetoolbox

makeatletter

%% https://tex.stackexchange.com/questions/487572/obeylines-and-gappto-from-etoolbox
%% Collect the body of the xltabular in tabbody:
begingroup
 lccode`~=`^^M
 lowercase%
endgroup
 longdeftabline#1~%
 xapptotabbodyunexpanded#1\hline%
 futureletnextlistit@checknext
 
 newcommand*listit@obeylinescatcode`~=active let~=tabline


newcommand*listit@checknext%
 ifxnextexplain
 letnext=relax
 else
 letnext=tabline
 fi
 next


NewDocumentEnvironmentlistit%
 gdeftabbody
 noindent
 beginminipagetextwidth
 raggedcolumns
 beginmulticols2
 col@number@ne
 mathchardefLT@end@pen=0 %
 begingroup
 listit@obeylines
%
 endmulticols
 endminipage

makeatother

NewDocumentCommandexplain%
 endgroup
 vspace-baselineskip
 beginxltabularlinewidthX
 hlinehline
 tabbody
 hlinenoalignvskip 4pt%
 endxltabular


begindocument

beginlistit
Lorem ipsum dolor
sit amet, consectetur
adipiscing elit
sed do eiusmod
tempor incididunt
bfseries See Note
ut labore
textbfSee Note
magna aliqua.
Paragraph break here:par But we remain in the same ``line.''
explain

Note: verb+bfseries+ is not ignored anymore. You can easily see that
verb+textbfSee Note+ works fine too.
endlistit

enddocument

Footnote

1. Such as a 〈letter〉, 〈otherchar〉 or unhbox coming from the expansion of leavevmode, among other possibilities (cf. TeXbook p. 283).