IMO...this truly depends on where you plan on taking the relationship or the expected depth of connection. If the nature of your relationship is one where you sustain essentially separate lives, or perhaps a more casual involvement, then it probably isn't necessary. If you are looking for more integration or want to share in more of the time of your partner then you'll probably need to at least be friendly.
Depth will be the key I believe. The idea of having a deep relationship without getting to know huge parts of your partner's life seems almost unobtainable to me. The unseen impact and influence of another person they love is a huge aspect of their lives. It will affect your life as well. I would much rather be comfortable with that aspect then live a life where you essentially pretend it does not exist.
If you are looking for a more "surface" relationship then I don't see the need to get overly involved with other aspects of their life.
My own relationship includes love and a deep level of caring/respect for both of Redpepper's other relationships. It wouldn't be healthy for me if it did not and would have a negative affect on all of us, just like if one of them were negative towards me. It inevitably would not be sustainable and seeing as we have long term goals in mind that would not work for us.
Hope this helps and I think you have a great attitude in wanting to embrace his other relationship