Redmine

Looks like you are missing some IMAP parameter.
You can see the list of all available parameters:
RedmineReceivingEmails

port=PORT                IMAP server port (default: 143)
ssl=SSL                  Use SSL? (default: false)
starttls=STARTTLS        Use STARTTLS? (default: false)

If IMAP over SSL/TLS is used, then probably you have to configure port=993 (default IMAPs port) and ssl=SSL.
If IMAP STARTTLS is used, then probably you have to configure port=143 (default) and starttls=STARTTLS.

We are using SSL, so I used this configuration

/usr/local/bin/rake -s -f /opt/redmine/current/Rakefile --trace redmine:email:receive_imap RAILS_ENV="production" host=mail.mycompany.com username=redmine@mycompany.com password=myPassword unknown_user=accept ssl=true port=993

I also tried with ssl=1, since I wasn't sure which values it really wants. I've seen examples of both.

With that, I get this error.

• Invoke redmine:email:receive_imap (first_time)
• Invoke environment (first_time)
• Execute environment
• Execute redmine:email:receive_imap
  rake aborted!
  OpenSSL::SSL::SSLError: SSL_connect SYSCALL returned=5 errno=0 state=unknown state
  /usr/lib/ruby/2.0.0/net/imap.rb:1454:in `connect'
  /usr/lib/ruby/2.0.0/net/imap.rb:1454:in `start_tls_session'
  /usr/lib/ruby/2.0.0/net/imap.rb:1047:in `initialize'
  /opt/redmine/current/lib/redmine/imap.rb:30:in `new'
  /opt/redmine/current/lib/redmine/imap.rb:30:in `check'
  /opt/redmine/current/lib/tasks/email.rake:116:in `block (4 levels) in <top (required)>'
  /opt/redmine/current/app/models/mailer.rb:432:in `with_synched_deliveries'
  /opt/redmine/current/lib/tasks/email.rake:115:in `block (3 levels) in <top (required)>'
  /var/lib/gems/2.0.0/gems/rake-10.4.2/lib/rake/task.rb:240:in `call'
  /var/lib/gems/2.0.0/gems/rake-10.4.2/lib/rake/task.rb:240:in `block in execute'
  /var/lib/gems/2.0.0/gems/rake-10.4.2/lib/rake/task.rb:235:in `each'
  /var/lib/gems/2.0.0/gems/rake-10.4.2/lib/rake/task.rb:235:in `execute'
  /var/lib/gems/2.0.0/gems/rake-10.4.2/lib/rake/task.rb:179:in `block in invoke_with_call_chain'
  /usr/lib/ruby/2.0.0/monitor.rb:211:in `mon_synchronize'
  /var/lib/gems/2.0.0/gems/rake-10.4.2/lib/rake/task.rb:172:in `invoke_with_call_chain'
  /var/lib/gems/2.0.0/gems/rake-10.4.2/lib/rake/task.rb:165:in `invoke'
  /var/lib/gems/2.0.0/gems/rake-10.4.2/lib/rake/application.rb:150:in `invoke_task'
  /var/lib/gems/2.0.0/gems/rake-10.4.2/lib/rake/application.rb:106:in `block (2 levels) in top_level'
  /var/lib/gems/2.0.0/gems/rake-10.4.2/lib/rake/application.rb:106:in `each'
  /var/lib/gems/2.0.0/gems/rake-10.4.2/lib/rake/application.rb:106:in `block in top_level'
  /var/lib/gems/2.0.0/gems/rake-10.4.2/lib/rake/application.rb:115:in `run_with_threads'
  /var/lib/gems/2.0.0/gems/rake-10.4.2/lib/rake/application.rb:100:in `top_level'
  /var/lib/gems/2.0.0/gems/rake-10.4.2/lib/rake/application.rb:78:in `block in run'
  /var/lib/gems/2.0.0/gems/rake-10.4.2/lib/rake/application.rb:176:in `standard_exception_handling'
  /var/lib/gems/2.0.0/gems/rake-10.4.2/lib/rake/application.rb:75:in `run'
  /var/lib/gems/2.0.0/gems/rake-10.4.2/bin/rake:33:in `<top (required)>'
  /usr/local/bin/rake:23:in `load'
  /usr/local/bin/rake:23:in `<main>'
  Tasks: TOP => redmine:email:receive_imap

Which version of OpenSSL is used on your Redmine server?

Try to connect to IMAP host using OpenSSL command:

openssl s_client -crlf -connect mail.mycompany.com:993

and examine output of command: server certificate, certificate chain, supported TLS/SSL protocol versions, and cipher.
If everything is working OK, you can issue several standard IMAP commands, e.g.:

a1 LOGIN redmine@mycompany.com MyPassword
a2 LIST "" "*" 
a3 EXAMINE INBOX
a4 FETCH 1 BODY[]
a5 LOGOUT