While "fast enough" is a relative term, and one might be satisfied while another is frustrated with any site's performance, I suspect the issue is more likely to be more related to the nature of "shared" hosting the way Dreamhost Does it than it is the "plan" you select ($8-$80 month).
In my experience, the type of site you describe might need to consider the following in determining Dreamhost's suitability for hosting it:
1) Depending on the efficiency of the PHP coding and the optimization of the MySQL queries (and how "big" your database is - you gave no specifics), the "extra trip" the Dreamhost shared ervers have to make to the seperate database machines (also shared) will introduce latency - how much depends on a ot of things (not the least of which are the number of users on each of those servers and what they are doing), but it should be considered.
2) If the site is completely dynamically generated (without serious caching) I can see 10,000 "highly interactive" visitors a day possibly causing problems with resource usage when sharing a server with others, depending upon what else is going on on your shared server and how your application(s)/queries are coded.
3) If your application has features such as automatically emailing updates, forum replies, etc. to users, you could easily have trouble with Dreamhost's newly implemented email limits (generally 200 per hour); it wouldn't take much at all to have your site generate more than that.
That said, bandwidth and server space should be more than adequate on even the $8.00 month plan; a more expensive plan will only give you more features, bandwidth, users, etc. but will have no effect on issues 1-3 mentioned above, as even those plans are still on shared servers.
The only other thought that comes to mind regarding your questions is that with 10,000 daily visitors you ought to be able to monetize the site sufficiently to afford a dedicated box, making many of these issues moot.
If not, or if you choose not to do so, you could certainly try it and avail yourself of the 97 day money-back guarantee if it does not work "fast" enough for you, or if the resource usage/email limits get to be problematic.