What are a tenant's rights when their landlord doesn't keep up with snow and ice removal?

Unless the lease says that the landlord will plow, he probably does not need to do so. That said, if anyone is injured because of snow or ice which the landlord knew or had reason to know of (e.g. he's aware it snowed) and did nothing about, the landlord would very likely be liable for their injury--so for their medical bills, missed work, etc. The threat of liability is normally what makes landlords clear the snow and ice; however, if your landlord is willing to leave with the threat of being sued and held economically responsible, you might not be otherwise able to make him him shovel, plow, etc. Unfortunately, in many cases, people can choose to be irresonsible and gamble that they won't suffer the consequences.