Cfrgtkky

We can use makebox[0pt]{text} to make a box that is logically 0pt width to external (as if nothing existed to the surroundings, while actually we can put whatever in it).
My questions is---what is the counterpart command to make the height of a box appear to be 0pt, logically, to its adjacent boxes.
With such a box, after a newline, next line will appear at the vertical position of the previous line on the same page.

To answer your written question: you can remove the height of a box by storing into a box register and set the height to 0pt and then use that box. You might also want to set the depth to 0pt. Ways of easily do this is the adjustbox package: adjustbox{set height=0pt, set depth=0pt}{Some Text} or, without a package, raisebox{0pt}[0pt][0pt]{Some Text}.

Note that this just makes the box have no official vertical size, but it is still placed in a horizontal list which makes up the line. Inside a paragraph all characters and other material like images are modeled by TeX as boxes with a height, depth and width and placed on a list which makes up a line, until enough is there to build a full line. All of these boxes sit on a line, the baseline, everything above that line makes up the height, everything below the depth. TeX usually places the amount baselineskip between the baselines of two sequential lines.

Therefore if you want to overprint the following line over the last line, deleting the official height and depth of the content of the first line has no effect, because the baselineskip is still added. In order to achieve this you need to undo the baselineskip again so that the second line starts at the same vertical position as the first. This can be done using vspace{-baselineskip} between the two lines, in most cases. Note that if the first line contains material with a very large depth, then the two lines will be more than baselineskip apart! Also if you have a paragraph break there then LaTeX might add more space between the last line of the last paragraph and the first line of the following paragraph depending on global settings (e.g. using the parskip package; used to get German paragraph style). Also LaTeX tries to spread the paragraphs across the page if flushbottom is active by increasing inter-paragraph spacing. In such cases, you might want to use \ instead of par.

documentclass{article}

usepackage{color}

newboxmybox

begin{document}

Some text pppp



Some text pppp



setboxmyboxvbox{% some arbitrary contents

   begingroupcolor{red}

   % some paragraphs

     hspace*{5cm}aaa, bbb, cccpar 

     hspace*{5cm}ppp, qqq, rrrpar

   % some vertical space

     vspace*{1cm}

   % some more paragraphs

     hspace*{5cm}uuu, vvv, wwwpar 

     hspace*{5cm}xxx, yyy, zzzpar

   endgroup}

% set the box to occupy zero space

htmybox0pt

dpmybox0pt

% insert it in a way keeping prevdepth

setboxmyboxvbox{boxmybox}

unvboxmybox



Some text pppp



Some text pppp

end{document}

Circumscribe suggestion: (if inserted contents are also paragraph like, hence one may want some alignment)

documentclass{article}

usepackage{color}

newboxmybox

newdimenmyprevdepth

begin{document}

Some text pppp



Some text pppp



myprevdepthprevdepth

setboxmyboxvbox{% some arbitrary contents

   prevdepthmyprevdepth

   begingroupcolor{red}

   % some paragraphs

     hspace*{5cm}aaa, bbb, cccpar 

     hspace*{5cm}ppp, qqq, rrrpar

   % some vertical space

     vspace{2baselineskip}

   % some more paragraphs

     hspace*{5cm}uuu, vvv, wwwpar

     hspace*{5cm}xxx, yyy, zzzdotfillpar

   endgroup}

% set the box to occupy zero space

htmybox0pt

dpmybox0pt

% insert it in a way keeping prevdepth

setboxmyboxvbox{boxmybox}

unvboxmybox



Some text ppppdotfill



Some text ppppdotfill



And more paragraphs...



mbox{}cleaders hbox to .44em{hss .hss }hskip5cm This is regular

textdotfillnewline which continues heredotfill



end{document}

I am adding a variant which avoids the unvbox trick and needs no handling of prevdepth.

documentclass{article}

usepackage{color}

newboxmybox

begin{document}parskip2pt

Some text pppp (parskip is 2pt in this example)



Some textrule{1cm}{1pt} pppp



setboxmyboxvtop{% some arbitrary contents

   hrule height 0pt

   vbox{}vskip-baselineskip

   begingroupcolor{red}

   % some paragraphs

     hspace*{5cm}aaa, bbb, cccpar 

     hspace*{5cm}ppp, qqq, rrrpar

   % some vertical space

     vspace{dimexpr2baselineskip+parskiprelax}

   % some more paragraphs

     hspace*{5cm}uuu, vvv, wwwpar

     hspace*{5cm}xxx, yyy, zzzdotfillpar

   endgroup}

% set the box to occupy zero space

htmybox0pt

dpmybox0pt

boxmybox % this will get the correct baseline

vskip-baselineskip



Some text ppppsmash{rule{1pt}{dimexprbaselineskip+parskip}}dotfill



Some text ppppdotfill



And more paragraphs...



mbox{}cleaders hbox to .44em{hss .hss }hskip5cm This is regular

textdotfillnewline which continues heredotfill



end{document}

This is essentially just a comment, but it doesn’t fit in 600 characters: is it this what you are trying to achieve?

% My standard header for TeX.SX answers:

documentclass[a4paper]{article} % To avoid confusion, let us explicitly 

                                 % declare the paper format.



usepackage[T1]{fontenc}         % Not always necessary, but recommended.

% End of standard header.  What follows pertains to the problem at hand.



newenvironment*{overlap}{%

    par

    null

    setlengthbaselineskip{0pt}%

    setlengthlineskip{0pt}%

    setlengthlineskiplimit{-maxdimen}%

    setlengthparskip{0pt}%

}{par}







begin{document}



This is too long for a comment: is it this what you expected?



begin{overlap}

    noindent A few\

    makebox[1cm]{}overlapping\

    makebox[3cm]{}lines.



    ..................................And another paragraph.

end{overlap}

More text.



end{document}

I forgot: this is the output.

The OP asked for a version of the above that doesn’t break down when the font size is changed; here it is:

% My standard header for TeX.SX answers:

documentclass[a4paper]{article} % To avoid confusion, let us explicitly 

                                 % declare the paper format.



usepackage[T1]{fontenc}         % Not always necessary, but recommended.

% End of standard header.  What follows pertains to the problem at hand.



usepackage{color}



newenvironment*{overlap}{%

    par

    null

    linespread{0}selectfont

    setlengthlineskip{0pt}%

    setlengthlineskiplimit{-maxdimen}%

    setlengthparskip{0pt}%

}{par}







begin{document}



This is too long for a comment: is it this what you expected?

begin{overlap}

    noindent A few\

    makebox[1cm]{}overlapping\

    makebox[3cm]{}lines.



    ..................................And another paragraph.

end{overlap}

More text.



Now it works also if you change the font size:

begin{overlap}

    noindent A few\

    Large makebox[1cm]{}overlapping\

    makebox[3cm]{}lines.



    color{cyan}Huge ..................................And another paragraph.

end{overlap}

Still more text.



end{document}

The corrseponding output:

I’m not (yet) going to provide explanations, because it is still unclear whether or not this is what the OP is trying to achieve.