For some reason, the "comments" function is not working and I'd like to make a few comments on John's thoughtful post.
Many patients who voice this kind concern with being a "burden" are, in my view, suffering from depression and I am quick to recommend these patients for consultation for depression.
However many are really saying that they do not want to suffer. They do not want to suffer loneliness or suffer dependence on others for their needs. They may fear physical pain.
It may be difficult, but if I know a patient is Catholic (and I will inquire about my patients' faith when they voice such a comment) it is useful to remind them that they can unite their suffering with that of Christ and share in His redemptive suffering. Salvifici Doloris, JPII's wonderful encyclical on suffering, is helpful in this regard (and well worth re-reading occasionally.)
I will generally also discuss why it may be important for the patient to "allow" the family to care for them. How this submission to dependency may be an opportunity to teach love to one's children and grandchildren. My mother, with her own eight children, welcomed her father and her mother-in-law into our modest home and provided to us, her children, an example of what it means to love. I am much richer because my grandparents "allowed" themselves to "burden" my mother. They were able to temper pride in order to become dependant.
I once had an old nun tell me that the engine of the Church, the spiritual energy of the Church, was the millions of old people continuously praying--praying the rosary, praying the Divine Chaplet, praying the hours, praying the daily Mass. Remind your patients that they can use this station in their life to enrich the Body of Christ through their prayers.