If it's on silent then it shouldn't make a noise or do you mean when you tap the keys because you can switch that off (settings>sounds>keyboard clicks) however i'm not sure if that takes the dialin noise off.
I turned off everything I could find including clicks and vibrate and I lowered the volume as low as it goes (no setting for zero bars) and the dial pad still makes noise when you press any key.
It's hard to believe Apple made no way to silence this.
You don't always need to talk. The most common reason would be calling an automated system where you don't need to speak or checking a remote voicemail etc..
A less likely situation would be in some kind of emergency where you were hiding and needed to quietly dial 911 without the the beep beep beep tones from the phone giving you away.
Unless you are blind, you don't need to hear these tones since you are looking at the phone and can see that the numbers you pressed went through correctly.
My previous cell phones had an option to have the keypad tones playing through your speaker on or off as needed. I don't know why the iPhone doesn't.
I just figured it out. I didn't understand what the person meant by the phone being "on silent."
There is no way to do it with menus or the volume up and down buttons, but if you use the physical switch on the left side of the phone and pull it away from the screen so red shows, then the phone is truly silenced including the touch tones.
This is the first time I even noticed that switch was there. I had to search online for how to silence the ring tone to find this, then I tested the keypad and it was also silent.
You know what? That's pretty darn funny! I thought about that while I was on the phone with the support guy, so I couldn't try it then. He really didn't seem to know the answer to my question, but told me that the silent button would not work, so I didn't even attempt it. I just tried it and you're right! So much for Apple Support!