Btukfyl

The Torah in many places speaks about tzara'at.

Is there any midrash that speaks about the details of the pain of someone infected with tzara'at?

I can't tell you how much pain a metzora would suffer, but a couple of Mefarshim indicate that they did experience some amount of pain.

Abarbanel to Vayikra 13:1 explains why Tzara'as is called a Nega ("plague"):

ונקראו אלו החליים נגע מפני שהיה העור נגוע בכאב

These illnesses are called "plagues" because the skin is plagued with pain.

Based on this understanding, the Abarbanel asks (on v. 47) on the concept of clothing receiving Tzara'as, as clothing can't feel pain:

ובהיות הבגד דבר שאין בו הרגש איך יתכן שיהיה בו צרעת ואיך יאמר עליו הכתוב צרעת ממארת הנגע טמא הוא כי לשון ממאר' מורה על הכאב כמו סילון ממאיר. ומי שאין לו הרגש כבגד וכעור אין בו כאב

And when it is clothing, something which cannot feel, how can it be that it receives Tzara'as, and how can the passuk say about it "Tzara'as 'mami'eres,' the plague – it is impure," for "mami'eres" speaks about the pain, as in the expression "prickling briers [silon mami'ir]"? For something which doesn't feel, like cloth and leather, it can't have pain!

Rashi to Vayikra 13:51 understands "mami'eres" in the same fashion:

צרעת ממארת. לְשׁוֹן סִלּוֹן מַמְאִיר (יחזקאל כ"ח), פויי"נט בְּלַעַז

Tzara'as mima'eres – an expression of "prickling briers," point in Old French.

Bamidbar Rabbah 7:4, in discussing the sin of the Eigel HaZahav, interprets Yeshaya 17:11 as indicating that they received Tzara'as (in the similar Midrash in Vayikra Rabbah 18:3, it says they received both Tzara'as and Zivah):

וּכְאֵב אָנוּשׁ, בּוֹאוּ וְקַבְּלוּ אֶת הַצָּרָעַת. וְלָמָּה קוֹרֵא אוֹתָהּ וּכְאֵב אָנוּשׁ, לְשׁוֹן גֶּבֶר, מִפְּנֵי שֶׁהִיא מַכָּה הַגְּבַרְתָּנִית. דָּבָר אַחֵר, וּכְאֵב אָנוּשׁ, שֶׁהִיא מְנַתֶּשֶׁת אֶת הַגּוּף שֶׁהִיא נִכְנֶסֶת בּוֹ

"And 'anush' pain" – come and receive the Tzara'as. And why does it call it "'anush' pain"? ['Anush'] means "man" [interpreting 'anush' as 'enosh'], for it is a powerful affliction. Another explanation of "and 'anush' pain": Because it weakens the body into which it enters ['anush' literally means 'acute'].

So it's very clear that there was definitely pain involved, though none of these sources are clear on how much. Abarbanel definitely learns that pain is an integral part of the Tzara'as.