How handicapped am I in graduate admissions if I graduated from a lower tier university?

You don't mention what field you're in. At least in the social sciences at my R1, we look at various factors. The admissions committee knows that people have to choose the college that they went to for various reasons other than just academic excellence.

One of the factors that I look at is trajectory. Someone who went to a public high school, then a community college, and then transferred to a public university, with perhaps middling grades the first year but then quickly ramps up to stellar grades by the time they graduate is very interesting to me -- much more than someone who got all As at a good school but doesn't seem like they pushed themselves very hard.

With the portfolio that you describe for yourself, I'd say you wouldn't be eliminated in the first round, but you'd struggle in the second and third rounds to stand out. How you stand out is up to you. I'd work on a stellar statement of purpose -- one that strongly articulates why you want to go to graduate school to study what you want to study.

Again, this is in the social sciences at my school (a large private R1), your mileage may vary.

Part of this is because brilliance by itself isn't enough for grad school in the social sciences. Perseverance and autonomy and the ability to get knocked down and get back up are also critical.

My impression after being accepted and discussing this with senior faculty was the following:

A large, top tier graduate program receives applications. Some fraction of these (say a quarter) are entirely unqualified and will be discarded.

Another, very small group are fantastically qualified, with great research experience and superb academic credentials from the very best schools. These few will almost definitely get in, but may not accept.

Finally, the largest group consists of people with credentials that are no better or worse than yours (maybe they attended a better school, but their record is slightly worse, or their recommenders are more prestigious, but the letters are less personal). If you are in this group, you should do everything you possibly can to improve your application, but ultimately you're playing the odds.

The biggest obstacle you will face at the top tier schools is that they receive so many more applications that the acceptance rate might only be a few percent. This means the odds are very unfavorable that any particular top tier school will accept you as a solid, but not extraordinary applicant.

This is how it was explained to me anyway. So, I don't think your school is a serious handicap, but it probably does exclude you from the "Oh my gosh, must accept and give all the money!" pile. You might consider applying to more programs to compensate.

EDIT: As Ben Webster pointed out, "Apply to more programs" is incomplete. You might consider applying to more high quality programs.

I recommend reading the following article:

Why You Can’t Catch Up, by Nancy Hass; The New York Times, (August 1, 2014)

The takeaway is that anything is possible, but you're much less likely to get into top tier grad schools with a low-tier undergraduate.