How to advance the tier, when you have maxed the weapon, and cannot perform other missions because..

This means you have to buy a new weapon for the tier you are on - usually a new sniper rifle each level/tier.

You can upgrade the weapon damage on a sniper rifle with blue battle tokens earned from the pvp arena that will get you through the first few missions of the next level/city/tier, but you will eventually be forced to buy a new sniper rifle, as you can't upgrade things such as the extra distance needed in later missions. Upgraded assault rifles with extra weapon damage can cover a few higher tiers, so at least you won't have to buy a new assault weapon every single level.