UPNP only works for the immediately adjacent router. So, when your gaming system attached to your ASUS router tells it to open a UPNP forward, only the ASUS "hears" the request. The FiOS router which is an additional hop upstream never hears it.
Now first, you don't absolutely need the FiOS router. If you only have internet, it's fine to use only the ASUS router as your primary router. I'd keep the FiOS router however as if you ever have issues, you'll need to install it in order for Vz to troubleshoot -- they won't deal with your own gear and it's hard to get past the first level tech support without their router in place. If you have FiOS TV however, you'll need to get a MoCA bridge so the STB's can continue to get their guide data and you'll lose the ability to remotely program the STB DVR, etc.
Another possible configuration which may work for you is to continue to use the double-NAT scenario you have with the ASUS router behind the FiOS gateway and then configuration the FiOS gateway to treat the ASUS router as the "DMZ host". You can configure the DMZ host on the FiOS router under the "Firewall" section "DMZ host". To do this, you will need to switch your ASUS router to use a static IP on it's WAN interface (so if your FiOS router is 192.168.1.1, use 192.168.1.2 as your static IP for the ASUS and specify your FiOS router as the gateway address). In this manner, the FiOS router will automatically port forward ALL traffic to your ASUS router's WAN interface and your ASUS router will control what gets port forwarded to your LAN network. This may not work all the time as some devices which use UPNP to configure a forward also sense the public interface address of the router and tell the remote service to use that IP address to reply -- which won't work because it will think the 192.168.1.2 address is the public address.
A bigger word of caution however -- why use UPNP at all? Any security expert will tell you that this is one of the worst ideas in networking ever invented. It essentially allows a local device to tell your firewall to open inbound traffic to it. Great for legitimate uses, but a nefarious program or piece of malware can also easily issue the same request and "poof" you've got an open conduit into your private network from the internet for this piece of nasty software to operate over.
I would recommend highly that you disable UPNP. Instead, manually port forward what ports are needed (this is easily researched online) and most gaming systems will automatically sense the lack of UPNP and adapt appropriately. You could still use the double NAT if you like with the ASUS behind the FiOS router and make the ASUS a DMZ host so that you need only program the port forward on the ASUS router.
Last option, use your ASUS router in "Access Point" mode. This turns off the router function in the ASUS but lets you still configure the Wifi and use it. In this configuration, you then hook the LAN side of the ASUS to a LAN port on your FiOS router (after configuring the ASUS with a static IP -- giving the LAN interface a static IP of something like 192.168.1.2). The FiOS gateway will still service all the DHCP and UPNP requests, but you'll be able to set and use the Wifi on the ASUS.