That depends on how it's implemented, but probably not. 3G phones still work. Even 2G phones still work. 5G networks will probably still do LTE, 3G and 2G. You'll need a new phone if you want 5G.
However:
1. There's no more spectrum space available. Opening up new band for 5G (which would be required) would put cellphones on satellite frequencies, and changing the frequency of a satellite isn't a simple thing like changing the frequency of a few thousand police radios (the last change) - it requires a space mission for each satellite. That's a significant part of the entire world economy in cost.
2. We already have 40+mbps downloads - on an internet that can't support more than about 2mbps data transfer over any bu the shortest distances. So you can do about 18 simultaneous downloads at a time at full speed. On a phone? Phones don't need more speed, all they need is lower latency, and that can be done by modifying 4G.
3. It wouldn't be done before about 2020 anyway - by which time you'd probably want to replace that aging S5 (or even aging S7 or Note 6).