What sort of problems have you encountered by outsourcing?

Maybe instead of problem list (that will be too long), my way to avoid problems: 
1. See outsourcing as means to help your business objectives (not an IT modernization) 
2. Cost is important element but is not alone, pick providers that have capabilities for transformation and can help you on your journey
3. Along the way, invest in many POCs as you can, even that you choose 1 project from 10, the scale benefits will weight a lot  

(Information Technology) Outsourcing (and managed services) can be a great productivity builder for an organization, especially if the organization's main business is not IT.  However, if the outsourcer is not properly monitored and in alignment with the organization as a whole, it can be very disruptive.  In my experience, in-house outsourcing works best when there is strong organizational oversight and direction for the outsourcer.

Vendor attrition causes significant quality issues and missing milestones

major communication issues.  poor delivery of service.  may be low cost but you better expect low service with it.

Communication and resource quality issues