Removing man pages on Ubuntu Docker installation

Short answer: no, this should not cause any major issue.

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TL;DR

I think that you will not cause any major damage except these two cases:

1. If do you need a manual for any command, you won't find it.
2. The /usr/share/man folder will grow when you do a package install/update.

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1. Disable the apt cache:

   When you install a package with apt-get or aptitude on a Debian-based system, the downloaded package is, by default, kept in the APT cache located at /var/cache/apt/archives. This is really not necessary as you typically do not re-install the same package ever again. Over time, the content in /var/cache/apt/archives will grow.

   • Create a file in /etc/apt/apt.conf.d/ called 02nocache with these contents:

     Dir::Cache "";
     Dir::Cache::archives "";
     

   • Clear the apt cache:

     sudo rm -rf /var/cache/apt/archives 
     

2. Disable man pages, locales and docs:

   You can disable a lot of rubbish doing this:

   • Create a file called 01_nodoc on /etc/dpkg/dpkg.cfg.d with these contents:

     # /etc/dpkg/dpkg.cfg.d/01_nodoc
     
     # Delete locales
     path-exclude=/usr/share/locale/*
     
     # Delete man pages
     path-exclude=/usr/share/man/*
     
     # Delete docs
     path-exclude=/usr/share/doc/*
     path-include=/usr/share/doc/*/copyright
     

   • Delete the current contents:

     sudo rm -rf /usr/share/doc/
     sudo rm -rf /usr/share/man/
     sudo rm -rf /usr/share/locale/
     

Logically there would be no harm for your system, but you still may loose when you need some man pages.

Also you should notice any update will create that directory again