This can be closed, it was bug in Dovecot which is fixed now:
http://hg.dovecot.org/dovecot-2.1/rev/d16e212531ec

Thank you.

For posterity sake, this is the dovecot thread:

http://markmail.org/message/dpq7qgqfym42ymfj

As I previously stated, this is looking very much like it has nothing 
to do with Horde/IMP.

Except I did read your message.  And I explained exactly what your 
**logs** showed.  Your logs show no messages in the first access.   
What else am I supposed to say?
Again, what can I tell you?  As my message demonstrated to you on the 
dovecot list, that version of dovecot is broken (at least with your 
configuration).

I stand by my belief that Dovecot is far superior to Courier.

Michael, you really SHOULD read people messages - based on what you 
said you didn't read my first message. My inbox had 7 unread messages 
all the time and the information about unread messages is appearing 
and disappearing just like i described in my first message.

I reported tons of real bugs so far and you were always telling me 
that Courier IMAP is broken and i should switch to Dovecot. Now, when 
i finally did it, IMP doesn't work even as good as in Courier. I'm 
using Dovecot from NEWEST Debian (Wheezy) and there's no easy way how 
to install and maintain different version of Dovecot in Debian.

Anyway, you are just telling that it's broken server without any 
investigation. I really heard this several times from you and then it 
turned out to be a bug in IMP.. So, PLEASE, can you, at least, read 
and try to understand my messages?

...but your first log has this:

C: 4 LIST () "" (INBOX INBOX.Karantena INBOX.Spam) RETURN (STATUS (UNSEEN))
S: * LIST () "." "INBOX"
S: * LIST () "." "INBOX.Karantena"
S: * STATUS "INBOX.Karantena" (UNSEEN 0)
S: * LIST () "." "INBOX.Spam"
S: * STATUS "INBOX.Spam" (UNSEEN 0)
S: 4 OK List completed.

Which indicates that INBOX has NO unread messages.  I don't care about 
messages that came in after the first log, because they are irrelevant 
to your bug report that unseen messages existed on the **first** access.

For the record, your IMAP server appears to be broken anyway.  The 
above response to the LIST-STATUS command is incorrect (it is missing 
an UNSEEN response for INBOX).  Considering that Dovecot stable of the 
2.1 branch is at 2.1.17, your 2.1.7 version is very old.  I would 
recommend upgrading.

That's NOT true and you can see it in the logs:

C: 4 SEARCH RETURN (MIN COUNT) UNSEEN
S: * ESEARCH (TAG "4") MIN 5702 COUNT 7

There you can see 7 unseen messages, which is correct. This part of 
communication happened when i whitched from different folder back to 
inbox and this is a time, when number of unseen messages is corrected 
in IMP (for a while).

According to that log, there was NEVER any unseen messages in that 
mailbox.  So I have no idea why you would see unseen messages appear.

You are going to need to track this down yourself, as I can't 
reproduce (and neither can anybody else).  Nothing changed in 6.1.4 
that would suddenly have caused this either.

Ou sh.t, sorry, i uploaded the wrong file! Here is the correct one.

Your first issue is that you are NOT connecting to your Dovecot server:

S: * OK [CAPABILITY IMAP4rev1 UIDPLUS CHILDREN NAMESPACE 
THREAD=ORDEREDSUBJECT THREAD=REFERENCES SORT QUOTA IDLE ACL ACL2=UNION 
STARTTLS] Courier-IMAP ready. Copyright 1998-2010 Double Precision, 
Inc.  See COPYING for distribution information.

Are you sure you don't have a broken proxy somewhere that is 
alternating accesses between the two servers?

Dovecot 2.1.7 (Debian Wheezy)

Attaching IMAP log, there are 3 parts:
(1) Communication when i clicked on 'Refresh' while there already was 
an incorrect unread messages information (=IMP was already showing no 
unread messages).

(2) Communication when i opened folder 'Rozpisane'.

(3) Communication when i again opened INBOX and information about 
unread messages was corrected (for a while).

Obviously, this is not reproducible.  At a minimum you will need to 
provide an IMAP log and or AJAX responses to even begin to debug.

Firefox 24.0, Kubuntu linux

Number of unread messages next to a folder (i tried only inbox) is 
disappearing. Always reproducable (dynamic view):
1.) See the number of unread messages next to the inbox.
2.) Click on 'Refresh' button and the number disappears like there are 
no unread messages.
3.) Open any other folder, then again open inbox and go to (1).

Problem stared to happen when we switched from Courier to Dovecot.