Is it permissible for me to perform sunnah prayers when I still have some obligatory prayers that are yet to be replaced?
Scholars have agreed that for those who purposely left or missed their prayers, then it is obligatory for them to qadha’ their prayers.  In madhhab Syafie, Imam al-Nawawi detailed this issue as the following: If a person leaves his prayer due to uzur syar’ie such as he fell asleep or forgot (not due to his negligence or because him taking it lightly), then it is sunnah for him to qadha’ his prayer immediately, however, if a person missed his prayer without any uzur syar’ie then it is obligatory for him to qadha’ his prayer. 
Whereas Ibn Hajar al-Haitami said: “For someone who missed his obligatory prayer – if the missed prayer is due to a debility – it is permissible for him to perform sunnah prayers whether it is rawatib sunnah prayers or others. If he missed his prayer without any debility, it is impermissible to perform any sunnah prayers before he first replaced his missed prayers. The reason is it is obligatory for him to replace the prayer immediately utilizing all of his time…”  This opinion of Ibn Hajar al-Haitami is also shared by Syeikh Zainuddin al-Malibari and it is explained it is prohibited to perform sunnah prayers if there are still obligatory prayers that he missed, but the prayer performed is still valid. 
It should be reminded that this prohibition is due to the fact that obligatory worship is more important and prioritized compared to sunnah worship. This is in line with His statement in a qudsi hadith which states:
وَمَا تَقَرَّبَ إِلَيَّ عَبْدِي بِشَيْءٍ أَحَبَّ إِلَيَّ مِمَّا افْتَرَضْتُ عَلَيْهِ
“And My servant does not draw near to Me with anything more loved to Me than the religious duties I have obligated upon him.” 
Furthermore, the “debt” with Allah SWT must be settled. Rasullullah PBUH said:
فَدَيْنُ اللَّهِ أَحَقُّ أَنْ يُقْضَى
“So, the debt due to Allah is the one which most deserves to be paid.” 
Although for those who always perform sunnah prayers such as the sunnah rawatib, Tahiyyatul Masjid, Dhuha and tahajjud prayer, for instance, maybe would very much like to continue with their common practice. Likewise, those who had just repented and turned back towards goodness would be motivated and would like to increase their worship.
Thus, there is a second opinion cited by muta’akhirin Syafi’iyyah scholars, al-‘Allamah Sayyid ‘Abd al-Rahman bin Muhammad bin Husain bin ‘Umar Ba ‘Alawi al-Hadhrami from Habib ‘Abdullah al-Haddad which explained, for the group of people who missed their prayers without uzur syar’ie, the obligation to qadha’ the prayers can be postponed (it isn’t commanded to be replaced immediately), appropriate with one’s capability and isn’t constricting, as long as it doesn’t become a form of negligence or taking it lightly. Habib ‘Abdullah al-Haddad support this opinion with the hadith:
بُعِثْتُ بِالْحَنِيفِيَّةِ السَّمْحَةِ
“I was commissioned with the straight and easy religion.” 
The Prophet PBUH said:
يَسِّرُوا وَلَا تُعَسِّرُوا
“Make things easy and do not make them difficult.” 
Al-‘Allamah Sayyid ‘Abd al-Rahman bin Muhammad bin Husain bin ‘Umar Ba ‘Alawi al-Hadhrami commented, saying, “This opinion, as you see prioritizes from what has been said by the fuqaha; regarding the obligation of striving one’s whole time to qadha’ other than the time needed for himself and family of whom it is obligatory for him to be responsible for, for the opinion would greatly burden him,” 
Another thing that should be considered is, sunnah prayers can be divided into two, which are zawat al-asbab and not zawat al-asbab. Our teacher Syeikh Muhammad bin Muhammad al-Mukhtar al-Syinqiti explained, sunnah prayers which are considered as zawat al-asbab are those of which some reasons occur before and not after it. He presented the opinion of Imam al-Nawawi and Ibn Qudamah in this matter. Sunnah prayers that have reasons for them (zawat al-asbab) such as sunnah tahiyyat al-masjid, eclipse, istisqa’, Eid prayers and others. Sunnah prayers which don’t have any reasons for them such as mutlak sunnah prayer. Hence, replacing the obligatory prayers are prioritized compared to sunnah prayers which aren’t considered as zawat al-asbab. As for sunnah zawat al-asbab prayers, they can be performed even when there are still obligatory prayers that have yet to be replaced.
To conclude, missed prayers are wajib (obligatory) to be replaced and prioritized. Sunnah worships shouldn’t be prioritized over qadha’ prayers which are an obligation. Concurrently, it is permissible to perform other sunnah prayers as stated in the fatwa issued by al-‘Allamah Sayyid ‘Abd al-Rahman bin Muhammad bin Husain bin ‘Umar Ba ‘Alawi al-Hadhrami on the condition that the facilitation doesn’t make a person takes the obligation of replacing his prayers lightly. Wallahu a’lam.