Many-year-old laptop batteries: safe in my closet? For how long?
Yes, laptop batteries are rigorously designed to be safe under all normal usage conditions, including long-term storage. If they are stored too long then they will self-discharge to voltages so low that the may no longer be safe to recharge. If this occurs the BMS (Battery Management System) will not allow any further use of the battery, and the OS will report that the battery needs to be replaced. 

When Li-ion cells are at low voltages (typically &lt; 1.5V) the copper current collector can dissolve. If the cell is later recharged to higher voltage then the copper will plate back out - but not necessarily in the same uniform way. In particular it may plate into sharp dendrites that could pierce the separator and lead to internal shorts and venting (possibly with flames). Note that this dangerous scenario can occur only if the battery is subsequently recharged (which is prevented by the BMS). In particular, it does not occur when the battery is self-discharging in long-term storage.

While there have been improvements such as ceramic separators to help control internal shorts, they cannot be completely prevented, so a well-designed BMS will always permanently disable the battery if it detects such dangerous conditions. 

However, for healthy batteries it usually takes a very long time for the cells to self-discharge to such low voltages (assuming they are stored at reasonable initial capacity, usually 40-60%). It is not unusual to hear of NOS (new old-stock) batteries over 5 years old that still work well. Occasionally one even hears of 10 year-old NOS batteries that still function (see e.g. many reports on Candlepowerforums or BudgetLightForum where users "harvest" 18650 cells from laptop packs for flashlights or DIY projects - where now we have the risky case where the user (vs. BMS) is responsible for deciding when the cell has discharged too low to be safe to recharge).