Does DNA polymerase I require a $3^\prime$ end?
DNA Pol I does require the 3' end of a previous nucleotide to initiate elongation. 

Regarding Okazaki fragments, this is accomplished by the annealing of small RNA primers to the lagging strand part of a replication fork. DNA Pol I extends the lagging strand off of the 3' end of those primers, generating the Okazaki fragments. The RNA primers are later removed, leaving gaps between Okazaki fragments, which are later filled in through the combined actions of DNA PolI and DNA ligase. 



In the case of telomeres, the final RNA primer can't be filled by DNA Pol I because polymerase requires a free 3' hydroxyl for the attachment of the first DNA nucleotide. After removing out the RNA primer, the first DNA nucleotide would need to be attached to the DNA nucleotide preceding it, which is not found in the case of the final primer on a telomere, since it's all the way at the end of the linear strand. 

In this case then, DNA Pol I removes the RNA primer is removed by DNA pol I but cannot replace it with DNA, leaving a gap (step 6 in the following image).