I'll probably have to work overtime every day next week.

Literal

Next-week [topic-は] every-day overtime-not-must [probably-でしょう].

残業 ('overtime work') + する = 残業する ('work overtime'). ~なければならない + でしょう combines obligation + conjecture — 'will probably have to do X.' Japan-specific work vocabulary; overtime is a significant cultural topic in labor discussions.