Not a ligature, unless someone has made a custom version of the font containing one. I'd say that's just T followed by dotless i, and if the latter needs kerning to look like this, then kerning has been applied and what you were told it incorrect.

(Dotless i character is “option-shift-b”.)
I checked the font, and the combination “Tı” is indeed kerned within the font, as John assumed.

The kerning given by the font is -80 (as in the first line below). To get the Tı as close as the jpg above, I have to fix it to about -115 in InDesign (as shown in the little box for kerning)

If the kerning was not done in InDesign, then I suspect a special kerning pair was added to the font.

It may also be possible with luaTeX to add that kerning pair without touching the font. That would satisfy, in some way, all the given requirements.

The sample appears to be a display version of Minion Pro Bold (notice the thin joins of the m), not the normal version from which Michel extracted the -80 value. The display could be kerned tighter, perhaps.

Or perhaps Optical Metrics is applied and the set size is very large. Not sure if the algorithm would kern it so tightly, but I wouldn’t put it past it.

I have the bold display of Minion version 1.011 and the kerning is tighter, -88. If I put the optical kerning with InDesign CS3, I get -69. That does not change if I choose 72 points or 96 points.

Maybe some other combination of font version+application. Or maybe that is it.

Added:

Here is a scaled link (with width=100%) to the original picture of magic2101 for rapid comparison.

The font is different, but there is no way to compare with the original (that picture was a reconstruction). It looks like bold subhead, and the kerning I get is -83

Not tight enough.

Is it the foundry font, or modified? I could get this without any kerning by making a modified font, having an extra dotless i with negative left sidebearing, and a {calt} feature...

Without modifying the font, and without using an explicit \kern command, it can be done with luaLaTeX using a little file, say Ti.fea with the lines

languagesystem DFLT dflt;
languagesystem latn dflt;
feature kern {
      pos \T \dotlessi -115;
} kern;

and then loading the font with the option [FeatureFile=Ti.fea]. That does not seem to interfere with the rest of the kerning.

Kerning can be edited in Quark XPress to be document-specific, and the values in one document may be exported to another.

Well, here is something for Quark to beat. I did not find a way to make a calt table for lualatex, nor to simulate adding a table that does not exist in a font. There is at least a salt table in Minion Pro so here is another experiment. This time, I used a little file I called salt.fea with the following lines

languagesystem DFLT dflt;
languagesystem latn dflt;

feature salt {
  script latn ;
    language dflt ;
       sub \i by \dotlessi ;
} salt ;
feature kern {
 script latn;
   language dflt ;
      pos \T \dotlessi -115;
} kern ;

I used it to define the fontface \head in the following luaLaTeX file

\documentclass{article}
\usepackage{fontspec}

\setromanfont{Minion Pro Bold Subh}
\newfontface\head[FeatureFile=salt.fea,
                  Style=Alternate]{Minion Pro Bold Subh}
\begin{document}
{\head Time} in Toronto
\end{document} 

which gave the following output:

Of course, the "font face" \head used to write "Time" is of little use but at least the word "Time" is properly coded, without cheating with an dotlessi in the input. The worst that can happen if salt.fea is used with a font that misses a salt table is that the output simply shows "Ti" instead of "Tı" which is exactly as should be.