Are there any types that shouldn't be in cold storage?
Depends on how you define "cold". Per the FAQ here most seeds do just fine stored in boxes in a dark closet as long as it is cool and dry. Light, heat and humidity are the dangers to seeds. Seeds that require cold stratification is a whole other matter and they are limited in number.
Many use the fridge, not because it is cold but because it is a consistent cool temps, mostly darkm and has low humidity.
Freezing seeds is normally for very long term storage not routine storage and its risk is once the seed is removed it needs to be planted ASAP as once it thaws it germinates. You can't take them in and out or re-freeze without loss of viability.
Read through he many FAQs here for more details.
Thanks for the info!