How handicapped am I in graduate admissions if I graduated from a lower tier university? Due to a combination of factors, I ended up going for my undergraduate studies to a relatively unknown university that was close to me geographically.

Let's say that I have a relatively standard background for someone trying to apply to top tier schools for graduate studies (some research experience, some low to medium quality publications, very strong grades, good recommendation letters). Then how much does the alma mater have an impact for someone like me? I obviously know that it is still possible to be accepted, but how much of a disadvantage am I looking at exactly?
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.