Time-irreversible stochastic processes are frequently used in natural sciences to explain non-equilibrium phenomena and to design efficient stochastic algorithms. Our main goal in this thesis is to analyse their dynamics by means of large deviation theory. We focus on processes that become deterministic in a certain limit, and characterize their fluctuations around that deterministic limit by Lagrangian rate functions. Our main techniques for establishing these characterizations rely on the connection between large deviations and Hamilton-Jacobi equations. We sketch this connection with examples in the introductory parts of this thesis. The second part of the thesis is devoted to irreversible processes that are motivated from molecular motors, Markov chain Monte Carlo (MCMC) methods and stochastic slow-fast systems. We characterize the asymptotic dynamics of molecular motors by Hamiltonians defined in terms of principal-eigenvalue problems. From our results about the zig-zag sampler used in MCMCs, we learn that maximal irreversibility corresponds to an optimal rate of convergence. In stochastic slow-fast systems, our main theoretical contributions are techniques to work with the variational formulas of Hamiltonians that one encounters in mean-field systems coupled to fast diffusions. In the final part of the thesis, we study a family of Fokker-Planck equations whose solutions become singular in a certain limit. The associated gradient-flow structures do not converge since the relative entropies diverge in the limit. To remedy this, we propose to work with a different variational formulation that takes fluxes into account, which is motivated by density-flux large deviations.