The Collaborative International Dictionary of English v.0.48:

Importune \Im`por*tune"\, v. i.
   To require; to demand. [Obs.]
   [1913 Webster]

         We shall write to you,
         As time and our concernings shall importune. --Shak.
   [1913 Webster]

The Collaborative International Dictionary of English v.0.48:

Importune \Im`por*tune"\ ([i^]m`p[o^]r*t[=u]n"), a. [F.
   importun, L. importunus; pref. im- not + a derivative from
   the root of portus harbor, importunus therefore orig.
   meaning, hard of access. See Port harbor, and cf.
   Importunate.]
   1. Inopportune; unseasonable. [Obs.]
      [1913 Webster]

   2. Troublesome; vexatious; persistent; urgent; hence,
      vexatious on account of untimely urgency or pertinacious
      solicitation. [Obs.]
      [1913 Webster]

            And their importune fates all satisfied. --Spenser.
      [1913 Webster]

            Of all other affections it [envy] is the most
            importune and continual.              --Bacon.
      [1913 Webster]

The Collaborative International Dictionary of English v.0.48:

Importune \Im`por*tune"\, v. t. [imp. & p. p. Importuned
   ([i^]m`p[o^]r*t[=u]nd"); p. pr. & vb. n. Importuning.]
   [From Importune, a.: cf. F. importuner.]
   1. To request or solicit, with urgency; to press with
      frequent, unreasonable, or troublesome application or
      pertinacity; hence, to tease; to irritate; to worry.
      [1913 Webster]

            Their ministers and residents here have perpetually
            importuned the court with unreasonable demands.
                                                  --Swift.
      [1913 Webster]

   2. To import; to signify. [Obs.] "It importunes death."
      --Spenser.
      [1913 Webster]

WordNet (r) 3.0 (2006):

importune
    v 1: beg persistently and urgently; "I importune you to help
         them" [syn: importune, insist]

Moby Thesaurus II by Grady Ward, 1.0:

84 Moby Thesaurus words for "importune":
   advocate, aggravating, annoying, appeal, apply pressure, approach,
   beg, beseech, beset, besiege, blandish, bothering, bothersome, bug,
   buttonhole, cajole, call on, call upon, coax, crave, disturbing,
   dun, entreat, exasperating, exert pressure, exhort, galling,
   harassing, high-pressure, implore, importunate, insist,
   insist upon, invoke, irking, irksome, irritating, jawbone, lobby,
   make a pass, make advances, make an overture, nag, nag at,
   overture, pesky, pester, pestering, pestiferous, pestilent,
   pestilential, plague, plaguesome, plaguey, plaguing, plead,
   plead with, ply, pray, press, pressure, proposition, provoking,
   push, recommend, soft-soap, solicit, supplicate, sweet-talk, tease,
   teasing, throw a pass, tiresome, tormenting, troublesome,
   troubling, urge, vexatious, vexing, wearisome, wheedle, work on,
   worrisome, worrying