> Also, does this happen with buffer text or on the mode line?  

I'm not sure about the mode line (buffer names with Tamil text render
just fine) but the header line can have clipped text.  The screenshots
are from a dired buffer.

> If it happens with buffer text, try these two experiments when it
> happens:
>
>   . move the cursor with C-f across the problematically-displayed
>     text, and see whether the display becomes correct and/or whether
>     you see some display artifacts, like "ghosts" of the cursor block
>     left behind;

I observe none of these.

>   . go to the problematically-displayed text and type "C-u C-x =",
>     then compare what you see with the results of "C-u C-x =" for
>     the same text when it is correctly displayed
>

The *Help* buffer also has incorrectly shaped text.

>> [2] Modifying the fontset in `server-after-make-frame-hook' does not
>>     make a difference.  I don't see the issue when I do not use the
>>     daemon but I believe I haven't run the non-daemon session long
>>     enough.
>
> Then please try running such a non-daemon session longer.  It is
> important to know whether this is at all related to daemon.
>

Yes, I will do and see if there are any changes.

> If it only happens with daemon sessions, I'll ask you to describe in
> more detail how you use those sessions.  In particular, do you use
> both GUI and TTY emacsclient frames in the same session, 

I only use GUI frames.  But I do use emacsclient -c --eval a fair bit in
scripts (that mostly launch a GUI frame and runs a command---`shell',
`org-capture', etc.).

> do you edit Tamil text in TTY frames, 

No.  I don't use TTY frames.  

> do you connect to the same server from different remote hosts, 

No such thing.

> or display frames on clients that use different font for Tamil?  

I'm not sure what you exactly mean here: all clients use the same font.

> Any other detail in your routine usage might give a clue.
>

Since I observed (clear-composition-cache) change the "size" of Tamil
text a tiny bit, I can say that I rely on text-scale-mode a lot.  I
increase the buffer text by 2 to 3 times when writing something as that
helps me focus a bit better.

> Thanks.