What reactions will their reaction rates increase with time?
Reactions that are autocatalytic can have their rate increase as a function of conversion.

One example is the decomposition of cellulose acetate by hydrolysis, also known as vinegar syndrome. The reaction is acid catalyzed, and it produces acid. In theory, the reaction gets faster. In practice, diffusion and limiting reagents many prevent a rate acceleration. Acetylsalicylic acid decomposes similarly. In the following equation, $\ce{R}$, could be the cellulose backbone or the salicylyl group. The acetyl group $\ce{-COCH3}$ is represented by $\ce{Ac}$. Thus acetic acid is $\ce{HOAc}$.

$$\ce{R-OAc + H2O -&gt;[\ce{H+ catalyst}] R-OH + HOAc}$$

Another example is the alpha halogenation of ketones or aldehydes, which is acid catalyzed and acid-producing.

$$\ce{CH3-CO-CH3 + Br2-&gt;[\ce{H+ catalyst}] CH3-CO-CH2Br + HBr}$$

A historically interesting example is the allotropic decomposition of tin, or "tin pest", in which tin metal transforms to a brittle form in cold weather. The book Napoleon's Buttons, which is about the influence of chemistry (wittingly or otherwise) on history, opens with the conjecture that tin pest is what really stopped Napoleon's invasion of Russia.

In addition, as this answer reminds us, the rate of the reverse reaction increases to a maximum during the establishment of dynamic equilibrium.