A weight of zero occurs if a record is out of universe. Both PANLWT and LNKFW1MWT are non-zero only for records who have data in two consecutive months. In addition, PANLWT is non-zero for records who are able to be linked backward one month (e.g. people who were interviewed in January who were also interviewed in December), whereas LNKFW1MWT is non-zero for records who are able to be linked forward one month (e.g. people who were interviewed in January who were also interviewed in February). The CPS 4-8-4 Rotation Pattern limits who can be linked using either weight according to their month-in-sample (MIS). For instance, a person interviewed in January who is in MIS 1 will have a value of zero for PANLWT because they were not interviewed in December; they will, however, have a non-zero value for LNKFW1MWT because they were interviewed in February (assuming they were not skipped or missed that month). So, for some individuals PANLWT is available and LNKFW1MWT is not available and for other individuals the reverse is true.
The appropriate weight to use depends on the design of the research (e.g. whether the analysis involves pooling multiple basic monthly samples or linking samples across individual months). Analyses in which you are pooling basic monthly samples (e.g. aggregating samples from multiple months or years in order to increase sample size in a cross-sectional dataset) should utilize WTFINL in order to generate population estimates. When pooling multiple basic monthly samples, you will want to be aware of changes in the universe across samples and make sure to divide the weights by the number of samples being pooled. Analyses that require linking the CPS (e.g. tracking individual-level transitions across months in order to create a panel dataset) should utilize PANLWT or an IPUMS-constructed longitudinal weight, depending on the research design.
In hopes of clearing up some confusion about these weights, I will describe how they are different, which is causing the zero values you mentioned. PANLWT (created by the Bureau of Labor Statistics) weights persons based on the population at time 2, whereas LNKFW1MWT (created by IPUMS) weights observations based on the target population at time 1. Analyses of individuals interviewed in February linked to their January interview for a backward-looking question (e.g., what fraction of people in February transitioned from unemployment in the previous month?) should use PANLWT (because the target population is focused on February, time 2). In contrast, analyses of individuals interviewed in January who are linked to their February interview for a forward-looking question (e.g., what fraction of people who were unemployed in January became employed in February?) is focused on a target population in time 1 (January) and should be weighted with LNKFW1MWT.
For more information on longitudinal weights, see the powerpoint presentations and lab exercises from the 2018 IPUMS CPS Linking Workshop and this page on Linking and the CPS.