Here are a few things to trouble shoot, but it will help more if you can tell us: when you press volume up and volume down, what is written on your screen?
Your phone may need a clock reset and a reboot: make sure your clock and date are correct. If not, connect to iTunes or to Wi-Fi to allow them to reset, then press & hold Power Button + Home Button together for ~10 seconds. Release and wait for phone to restart (or restart it).
If this does not help, you could have displaced something during the repair:
1/ Torn flat cable that controls Volume Up/Down and Mute: remove the back-cover of your iPhone. Look at the top right side of the battery. You will see the flat cable running there. Check if it is broken/torn in anyway.
2/ This cable connects to the board under the big top plate. Remove that plate and disconnect the cable then re-seat it. Is sound restored? If not, disconnect the connector and try again while disconnected.
3/ Examine the area around the battery connector. Is something on the board dislodged/missing/broken?
4/ The charging dock connects to the logic board in the middle, next to the battery connectors, and is covered with a plate held down by 2 screws. Remove the plate, and re-seat the connector.
See if the sound is restored. If not, disconnect the connector, and try again while disconnected.
5/ While prying to remove the battery, did you squeeze on the logic board? You could have broken something.
Let us know if any of the above helps, otherwise we'll see what else you need to look at.