Is it necessary to move from the place one performed the obligatory prayer to perform sunnah rawatib prayer?
It is sunnah to change or move places to perform the sunnah rawatib prayer so that it will not be the same place as where one performed his obligatory prayer. This is in accordance with the advice of Muawiyah radhiyallahu ‘anhu to al-Sa’ib Ibn Yazid:
إِذَا صَلَّيْتَ الْجُمُعَةَ، فَلَا تَصِلْهَا بِصَلَاةٍ حَتَّى تَكَلَّمَ أَوْ تَخْرُجَ، فَإِنَّ رَسُولَ اللهِ صَلَّى اللهُ عَلَيْهِ وَسَلَّمَ أَمَرَنَا بِذَلِكَ، أَنْ لَا تُوصَلَ صَلَاةٌ بِصَلَاةٍ حَتَّى نَتَكَلَّمَ أَوْ نَخْرُجَ
“If you offer the Friday prayer, do not connect it with another prayer, till you talk or go out. Allah’s Messenger (ﷺ) had commanded us accordingly: That we should not connect a prayer to another prayer till we talk or go out (in between them).” 
Imam al-Nawawi said: “This is the evidence stated by al-Syafi’iyyah scholars saying that before one performs the sunnah rawatib prayer or other sunnah prayers then he should move to another place, a place different from the place he performed the obligatory prayer. The most afdhal is to perform it at home. If not, it should be performed at another replace in the mosque or at another place altogether with the purpose of increasing the places of prostration. The reason is to separate the performance of the obligatory prayer and sunnah prayer. He said, ‘until we talk’ is the evidence that the separation between the obligatory prayer and sunnah is achieved even just by talking. However, moving from the place is prioritized as we have previously stated.” 
Increasing the places of prostration is so that we hope the places where we perform worship will act as our witness for the people who prostrated as stated by Allah SWT:
يَوْمَئِذٍ تُحَدِّثُ أَخْبَارَهَا
“That Day, it will report its news.”