For the radiation treatment of lung cancer patients, four-dimensional computed tomography (4D-CT) is a common practice used clinically to image tumor motion and subsequently determine the internal target volume (ITV) from the maximum intensity projection (MIP) images. ITV, which is derived from short pre-treatment 4D-CT scan (<6 s per couch position), may not adequately cover the extent of tumor motion during the treatment, particularly for patients that exhibit a large respiratory variability. Inaccurate tumor localization may result in under-dosage of the tumor or over-dosage of the surrounding tissues. The purpose of this study is therefore to assess the degree of tumor under-dosage in case of regular and irregular breathing for proton radiotherapy using ITV-based treatment planning. We place a spherical lesion into a modified XCAT phantom that is also capable of producing 4D images based on irregular breathing, and move the tumor according to real tumor motion data, which is acquired over multiple days by tracking gold fiducial markers implanted into the lung tumors of patients. We derive ITVs by taking the union of all tumor positions during 6 s of tumor motion in the phantom using the first day patient tumor tracking data. This is equivalent to ITVs generated clinically from cine-mode 4D-CT MIP images. The treatment plans created for different ITVs are then implemented on dynamic phantoms with tumor motion governed by real tumor tracking data from consecutive days. By comparing gross tumor volume dose distribution on days of 'treatment' with the ITV dose distribution, we evaluate the deviation of the actually delivered dose from the predicted dose. Our results have shown that the proton treatment planning on ITV derived from pre-treatment cine-mode 4D-CT can result in under-dosage (dose covering 95% of volume) of the tumor by up to 25.7% over 3 min of treatment for the patient with irregular respiratory motion. Tumor under-dosage is less significant for the patient with relatively regular breathing. We have demonstrated that proton therapy using the pre-treatment 4D-CT based ITV method can lead to significant under-dosage of the tumor, highlighting the need for daily customization to generate a target volume that represents tumor positions during the treatment more accurately.