When you submit application for a long term Schengen Visa (D-visa) or resident permit extension or renewal you will receive a stamp in your passport indicating that the application is currently in process. This stamp allows you to REMAIN in the country, but DOES NOT ALLOW YOU TO LEAVE AND RE-ENTER.
Because there are no borders within the Schengen Zone, it is possible to travel around the zone while application is in process, but you cannot leave.
If you apply for a Schengen visa in another country, the pending application will be canceled as it will be considered no longer necessary. DO NOT DO THIS.
The only exceptions could be an emergency (medical or death of immediate family member, etc.) that would allow ability to petition for a short term visa while application renewal is in progress. Do not try to do this if it is not a real emergency. You will need to show written statement from doctor/hospital or death certificate. If you cannot provide this, it would be considered false pretense, pending application will be refused, and you could be barred from Schengen Zone for up to 10 years.
TL;DR – If your renewal application is currently pending, do not travel outside Schengen Zone.