No. Toothpaste is one of those things that in most modern societies we like to think we cannot do without. Not true. Mostly, just brushing is all that is necessary. Followed by rinsing. 99.99% of the effects of toothpaste are to get you to salivate while brushing and to make the taste of semi-spoiled food un-noticeable. A sip of water will do close to the same. That said, I will also note that saliva is as close to a universal solvent you will likely find.
Most tooth pastes have some sort of polishing compound, read abrasive. It will scrape the top layer (usually one or two molecules worth) off your teeth. I tend to not use that much, anyway. But this can lead to a reduction in enamels after many brushings.
Fluoride can harden your teeth. Good for those with soft teeth. Alcohol (ethyl) can can kill bacteria pretty well, good for those with gingivitis. Needed? Probably not on the trail. Nor is it needed for a week or around a month. For the long distance hiker, it helps to have a fluoride tablet (1/8 of a child's dose is enough for an adult, possibly too much. Every other day?) For those with gingivitis, splashing a teaspoon of fuel to cause salivation and kill bacteria, while brushing, is probably not a bad idea, even at home. Toothpaste is not ever necessary, though.
Brushing is important. Toothpaste may or may not be. On the trail, it can be skipped. You can train yourself to salivate. No, you are not Pavlov's experiment. Just close your eyes and think of something that really makes your mouth water as you bite into it. After a few times, you will get the trick.