The more apps you have running yes....
iPhone apps are a few MB, 20MB max. I would suspect you could download every single app out now and still have plenty of space on the 16GB iPhone.
Edit:
I did some math- there are currently 713 apps for download. At 20MB each, that's just over 14GB. I currently have 16 apps on my iPhone, from games to AIM. Total space required for those apps comes to 23.5MB. That's just over 1.5MB per app. If that's the average, the total space required for all apps is just over 1GB. It's a far cry from filling the HD on the 16GB iPhone and "slowing" it down with too much information.
I've owned an iPhone for a year now, mid July '07. I can speak from experience here when I say slow downs are normal. I was disappointed when I had to wait when switching from surfing to texting but it's normal. It's a great phone but it's not perfect. I do wish it was faster and that I could listen to music, surf, and occasionally answer a text all at the same time but it's just not possible on a PHONE!
If you think having 10 apps on the phone instead of 20 may possibly save you from waiting a few extra seconds for the buffer to catch up to your typing speed, then only download 10 and keep a few GB memory free. I, on the other hand paid for 8GB, and intend to use every single kb allowed without regret.